The Role of Capital Structure in Cross-Sectional Tests of Equity Returns

The Role of Capital Structure in Cross-Sectional Tests of Equity Returns Anchada Charoenrook This version: January, 2004 I would like to thank Joshua D. Coval, Wayne E. Ferson, William N. Goetzmann, Eric Ghysels, David Hirshleifer, Phil Howrey, Gautam Kaul, Craig Lewis, Ronald Masulis, Tyler Shumway, Douglas Skinner, Richard Shockley, Hans Stoll, and Anjan V. Thakor for helpful comments. All errors are the responsibility of the author. The Owen Graduate School of Management, Vanderbilt University, 401 21st. Avenue South, Nashville TN 37215. Phone: (615) 322-1890 Fax: (615) 343-7177 Email: anchada.charoenrook@owen.vanderbilt.edu 1

The Role of Capital Structure in Cross-Sectional Tests of Equity Returns Abstract: This paper examines the impact of time varying capital structure in cross-sectional tests of equity returns (the capital structure effect). Theoretical analysis yields very different empirical implications and interpretations than is common in the asset pricing literature. The analysis shows that the capital structure effect biases regression coefficients in the Fama and MacBeth (1973) estimation, and can induce the relation between size or book-to-market and equity returns. A new empirical test that distinguishes this explanation from other explanations in the literature reveals that in a cross section of equity returns the capital structure effect accounts for at least 50% of the explanatory power of size and 70% of the explanatory power of book-to-market.

1 Introduction Over the last two decades, empirical studies applying the Fama and MacBeth (1973) cross-sectional test of average returns report consistent evidence that size and book-to-market (BE/ME) are related to average equity returns, while the Capital Asset Pricing Model (CAPM) beta is not (See, for example, Banz (1981), Fama and French (1992), Fant and Peterson (1995), Kothari, Shanken, and Sloan (1995), Jaganathan and Wang (1996), Knez and Ready (1997), and Davis, Fama, and French (1999)). The interpretation of these findings is central to understanding what risks determine expected stock return. In this study, I reexamine this evidence recognizing that financial leverage, which differs dramatically across firms and is time varying, can have an important impact on stock return patterns. This analysis leads to a reassessment of implications of a large body of empirical work testing the determinants of expected stock returns. A number of interpretations of the size and book-to-market evidence have been offered in the literature. One line of thinking is that expected equity return is determined by a multifactor model. Size and BE/ME proxy for factor loadings on macroeconomic risks that are not captured by the Sharpe Lintner CAPM beta. For example, Chan and Chen (1991), Fama and French (1992, 1995, 1996), and Berk, Green, and Naik (1999) suggest that size and BE/ME proxy for a factor loading on financial distress risk that varies inversely with business cycle. Lettau and Ludvigson (2001) report that the log of consumption-wealth ratio, which captures economic cycles, performs as well as size and BE/ME in explaining the cross-section of equity returns. Berk (1995) points out that there is an intrinsic relationship between the market values of stocks and expected returns. Given relatively stable dividend policy, a rise in future expected returns requires current stock price to decline. Thus, any variable that includes a measure of price such as size or BE/ME is related to expected return when a risk factor is unaccounted for in asset pricing model estimates. Other researchers argue that parameter estimates of size and BE/ME are biased due to an errors in variables problem in the CAPM beta estimates that attenuate their statistical significance. 1 In recent work Ferguson and Shockley (2003) show that there is an errors in variables problem due to estimating a CAPM beta using market equity returns rather than including both equity and debt 1 For example, Kim (1995) shows that correcting for an errors in variables problem in the CAPM beta diminishes the size effect. 1

returns in the market index, which is theoretically more accurate. This creates a bias in the size and BE/ME parameter estimates. On the other hand, a number of researchers have challenged the validity of efficient market hypothesis arguing that conditional on the validity of conventional specifications of the asset pricing model, size and BE/ME are firm characteristics unrelated to factor loadings. These researchers conclude that size and BE/ME are related to future returns of stocks because market participants misprice stocks. That is, small size and high BE/ME firms are undervalued with respect to their fundamental risks. 2 Despite considerable debate over the interpretation of size and BE/ME evidence and the appropriate specification for testing the CAPM, to date the literature has come to no consensus. This study proposes an alternative explanation for the size and BE/ME effect, and presents new supporting empirical evidence. The explanation is based on a fundamental observation made by Black and Scholes (1973) in their seminal paper on option pricing. In any linear factor model, each factor loading on a levered firm consists of a factor loading on the firm assets (the unlevered firm) multiplied by the elasticity of equity price with respect to the underlying asset price (henceforth referred to as the price elasticity). 3 A factor loading on firm asset returns measures exposure of firm assets to an economic risk, while the price elasticity measures the added risk exposure due to financial leverage. In the context of the CAPM, equity beta is a product of the firm s asset beta multiplied by an elasticity term, which is a non-linear function of the firm s debt-to-equity ratio, measured in market value terms. Empirical tests here reveal that size and BE/ME are correlated with the debt-to-equity ratio, and thus they are also correlated with price elasticity. The hypothesis that size and BE/ME are related to expected equity returns because they proxy for price elasticity, rather than for firm exposure to a particular risk factor is formally tested in this study. alternative explanation is termed the capital structure effect. This paper provides two fundamental contributions. First, it provides a theory that explicitly shows how the capital structure effect can induce a relationship between size or BE/ME and 2 See for example, Lakonishok, Shleifer, and Vishny (1994), Daniel and Titman (1997), Daniel, Hirshleifer, and Subrahmanyam (1998). 3 For a detail discussion on this topic see Hamada (1971) for the case of riskless debt and Galai and Masulis (1976) for debt with default risk. This 2

expected equity returns. The theory developed here has several new empirical implications. 4 First, if the asset pricing model estimated is missing any risk factor, size and BE/ME will be related to expected equity returns because they proxy for the price elasticity component of the missing factor. Second, even if an economist identifies every risk factor that is priced, in cross sectional analysis size and BE/ME will still be related to expected equity returns. This is because estimation of a conditional factor loading in a Fama-MacBeth framework assumes that a factor loading is constant for a period of five years. Capital structure, however, varies continually over this five-year period. When firms keep their debt levels relatively constant and stock prices change, a systematic error, which is correlated with the debt-to-equity ratio, is introduced in the factor loading estimates. The error biases the coefficient estimates of factor loadings tested and of size and BE/ME. 5 This theoretical result highlights a problem in current test procedures. The coefficient estimate of any factor loading which is obtained by estimating the Fama-MacBeth first pass regression using levered equity returns is biased. Whether the bias is empirically significant or not is an empirical question which is explored here. Third, the relations between stocks and size or BE/ME are stronger for firms exhibiting higher leverage, everything else equal. 6 empirically important in explaining expected firm returns. Finally, size and BE/ME should not be Differentiating the capital structure effect from a distress risk factor effect is an important issue. It can resolve whether there is a distinct bankruptcy or financial distress risk factor related to economic cycles that causes a shift in the investment opportunity set of the marginal investor. If size and BE/ME proxy for a price elasticity and not for additional fundamental risk factors, the conventional use of factors SMB and HML as linear risk factors to evaluate the systematic risk of equity is brought into question. 7 Furthermore, if size and BE/ME proxy for the stock return elasticity, their relationships to equity returns is not counter factual evidence on the empirical validity of the conventional specification of the CAPM or of market efficiency. Established empirical methods test the effect of leverage by incorporating an additional leverage proxy term in a linear asset pricing model. 8 This method suffers from a well known fundamental 4 See Black and Scholes (1973), Merton (1974), and Galai and Masulis (1976). 5 Baker and Wurgler (2002) find that firms try to time the market when issuing debt, and firms keep debt levels constant for a long period of time. 6 This is consistent with empirical evidence reported in Knez and Ready (1997). 7 SMB and HML are factors created from portfolios of size and BE/ME proposed by Fama and French (1995). 8 See, for example, Bhandari (1988) and Fama and French (1992) 3

drawback; it concurrently tests multiple hypotheses. Furthermore, the elasticity term is a nonlinear function of the debt-to-equity ratio, and it multiplies firm-level factor loadings. Due to non-linearity, accounting for the capital structure effect by adding a leverage measure in a linear asset pricing model is ineffective. 9 The second contribution of this study is to develop an empirical test which distinguishes between the capital structure effect and other explanations. This test exploits the theoretical implication that size and BE/ME should not be empirically important in explaining expected firm returns. Comparing Fama-MacBeth estimates of equity and firm return factor loadings yields a method for testing the capital structure effect. For example, if size is a proxy for a distress factor, then size should explain the cross-section of both firm returns and equity returns, and the price of risk should be the same in both cases. Similarly, there should be an intrinsic relationship between size and firm returns if a risk factor is missing. Alternatively, if markets are inefficient with respect to equity returns, they should be inefficient with respect to firm returns as well. But if capital structure is the cause for equity returns being related to size or BE/ME there should be no relations between firm returns and either size or BE/ME. Empirical tests presented here are based on monthly equity returns and firm returns of NYSE, Nasdaq, and AMEX listed companies over the period from June 1963 through July 1998. Firm returns are estimated from weighted average of equity and debt returns. Comparisons between tests of firm returns and equity returns for the same set of firms demonstrate that in a cross section of equity returns at least 50% of the explanatory power of size and 70% of the explanatory power of BE/ME are due to the capital structure effect. 10 9 This may explain why some empirical studies such as Bhandari (1988) and Fama and French (1992), find that adding a leverage proxy as another linear term in the Fama-MacBeth test does not change the cross-sectional coefficients of size or BE/ME. 10 In a contemporary work, Hecht (2002) empirically tests equity, firm, and bond returns to support or reject a model of rational equilibrium, and Hectch reports similar empirical findings on BE/ME. My work differs from his study in two respects. First, I analyze the effect of a time-varying capital structure on the Fama-MacBeth test of equity returns, and I use firm returns in testing the theory. Second, my data set is different. Hecht studies the period from 1973 through 1997 with an average of approximately 450 observations per month; the data set used here has an average of 2700 observations per month. The longer period allows for an examination of the relationship between size and expected firm returns over varying market conditions; this relationship is not statistically significant after the 1970s, which is the shorter period that Hecth studies. A larger sample size also improves the power of the tests here. 4

2 The capital structure effect The theoretical analysis of the capital structure effect proceeds in two steps. The equity return process of a firm is derived from a return on its assets that is assumed to be lognormal. Then two scenarios of the Fama and MacBeth (1973) (FM) test are analyzed using this levered equity return process. Assume there are two orthogonal marketwide variations denoted by W l,t, for l [1, 2]. 11 Let the asset return follow a lognormal process. Under widely accepted market conditions and without assumptions about a specific capital structure policy of a firm, the excess equity return process of firm i at time t, r i,t, is given as r i,t = ν i,t βi1,t λ 1,t + ν i,t βi2,t λ 2,t + 2 l=1 ν i,t σ il,t dw l,t + ν i,t σ i,t dɛ i,t. (1) All derivations are provided in the appendix. The parameter σ il,t is the conditional covariance of the asset return with marketwide variations, and σ i,t is the conditional firm-specific variation. They are non-stochastic. Both dw l,t and dɛ i,t are standard Brownian motion processes and E[dW l,t dɛ i,t ] = 0. A line over a parameter denotes a parameter associated with firm value. The variable β il,t is the firm-level factor loading defined as β il,t Cov( r i,t, W l,t )/V ar(w l,t ), where r i,t is the return on the assets of the firm. The variable ν i,t is the elasticity of the stock price with respect to the value of the assets of the firm defined as ν i,t S i,t V i,t V i,t S i,t, where S i,t and V i,t denote equity price and firm value of firm i at time t. The price elasticity ν i,t is a constant one for unlevered firms. It is a function of the debt-to-equity ratio for levered firms, and under general assumptions it is an increasing function of the debt-to-equity ratio. 12 The equity return process in (1) illustrates that the conditional equity beta, the correlation of equity return with marketwide variations, and firm-specific volatility equal the respective firm parameters multiplied by ν i,t. Therefore these equity level parameters increase with a firm s leverage. 11 The appendix derives the equity return process for any arbitrary number of marketwide variations. 12 It is shown in the appendix that, under the assumptions of the Merton (1974) capital structure model, price elasticity increases with the debt-to-equity ratio. 5

This is consistent with existing empirical evidence. 13 Harvey and Siddique (2000) find that the conditional skewness of equity returns declines with size but increases with BE/ME. According to Equation (1), when there is a negative shock to returns, leverage increases, and ν i,t increases, causing further increases in volatility. This interaction generates more negative skewness in the probability distribution of equity returns of firms with higher debt-to-equity ratios. The empirical test results presented below reveal that size is negatively correlated and BE/ME is positively correlated with the debt-to-equity ratio. Thus, leverage may explain why size and BE/ME forecast conditional skewness in the cross-section of equity returns. 2.1 The Fama-MacBeth cross-sectional test of levered equity returns The Fama-MacBeth test and its variants consist of two regressions. 14 The first-pass regression estimates the CAPM equity beta by estimating the market model using three to five years of historical monthly return data, assuming that beta is constant. The CAPM beta estimate is then employed as an independent variable in the second-pass regression. The second-pass regression is a time series of cross-sectional ordinary least squares or weighted least squares regressions of excess returns, r i,t, on one or more independent variables, X il,t, that are candidates for factor loadings. For each test period, from t = 1,..., T, and for N independent variables the regression is: r i,t = ˆλ 0,t + N ˆλ l,t X il,t + ξ i,t, (2) l=1 where ξ i,t is the residual error term. The hat symbol denotes an estimated variable. The time series average of the cross-sectional regression coefficients of the factors is estimated from ˆλ l = 1 T Tt=1 ˆλl,t. Each average slope is tested against the hypothesis that it is zero to identify which risk factors determine expected returns. When levered equity returns are involved in the Fama-MacBeth test, I show that size and BE/ME may explain average returns even though they are unrelated to any risk factors. That is, 13 Christie (1982), for example, reports that the variance of equity returns increases with the level of leverage. DeJong and Collins (1985) find that firms with higher leverage have more unstable equity beta than firms with less leverage. 14 For a comprehensive review of the two-pass test of beta models, see Shanken (1992). 6

size and BE/ME are independent of dw 1,t, dw 2,t, β i1,t, or β i2,t. 2.2 If the asset-pricing model is missing a risk factor Suppose we test an asset pricing model that is missing one factor from what would be the true model, but that includes either size or BE/ME (denoted here by X i,t as one of the independent variable). The Fama-MacBeth cross-sectional regression is given as r i,t = ˆλ 0,t + ˆλ 1,t β i1,t + ˆλ X,t X i,t + ξ i,t. (3) When this regression is estimated using levered equity returns from Equation (1) the time series average of the cross-sectional regression coefficient of size or BE/ME is ˆλ X = E t [ˆλ X,t ] = E t[e i [ β i2,t ]]E t [λ 2,t ]E t [Cov i (ν i,t, X i,t )]. (4) Var i (X i,t ) The Cov i (, ) denotes the covariance in the cross-section of firms. Since the time series and crosssectional average of firm-level factor loading E t [E i [ β i2,t ]] > 0, and the risk premium of the missing factor E t [λ 2,t ] > 0, the average slope of size or BE/ME, ˆλ X, is not zero but has the same sign as the correlation between size or BE/ME and ν i,t. The derivation of Equation (4) assumes that neither size nor BE/ME is related to any economic risk factor. Yet size and BE/ME explain the cross-section of average equity returns. The intuition for this result is that size or BE/ME proxies for ν i,t which is a component of the equity-level factor loading of the missing factor. [Insert Figure 1 here] Is size or BE/ME correlated with ν i,t? Although ν i,t is unobserved, it increases with the debt-toequity ratio of a firm. Therefore, if size or BE/ME is correlated with the debt-to-equity ratio, it is correlated with ν i,t as well. Figure 1a reports the time series of monthly cross-sectional correlations between size and the debt-to-equity ratio and BE/ME and the debt-to-equity ratio from July 1967 through June 1998. The correlation between size and the debt-to-equity ratio is negative in 96.7% 7

of the months and the time series average of these correlations is -0.07, which is significant at a 1% level (Table 1). This correlation is negative and higher in magnitude during all months before 1972. The correlation between BE/ME and the debt-to-equity ratio is positive in all months (second plot in Figure 1a). The time series average of this correlation is 0.258, which is significant at a 1% level. [Insert Table 1 here] The conditional price elasticity ν i,t can be calculated from firm characteristics assuming a specific capital structure model. Using the Merton (1974) capital structure model and data described below I estimate the conditional price elasticity for each firm each month. The monthly cross-sectional correlation of size and ν i,t is negative in most months, and the correlation of BE/ME and ν i,t is positive in all months (Figure 1b). The time series average of the correlation of size and ν i,t is -0.064 and that of BE/ME and ν i,t is 0.259 (Table 1). These empirical findings suggest a capital structure effect as follows. The average slope coefficient of size in the Fama-MacBeth test is negative, and it is steeper prior to 1972. This may explain the empirical puzzle of why size does not explain the cross-section of equity returns after the early 1970s. The capital structure effect predicts that the average slope of BE/ME is positive. Furthermore, since BE/ME and the debt-to-equity ratio or ν i,t are more correlated than size and the debt-to-equity ratio or ν i,t, Equation (4) predicts that the average slope of BE/ME will be steeper than the average slope of size. This is consistent with the empirical evidence reported in Fama and French (1992 and 1996). Finally, since ν i,t is constant for firms, size and BE/ME should not explain the cross-section of average firm returns. 2.3 If a factor loading is estimated from Fama-MacBeth first pass regression When a factor loading such as the CAPM beta is unobservable, it is obtained from the first-pass regression of the Fama-MacBeth test by estimating the market model using returns for 60 months immediately prior to the test period. Given the time variation of leverage during this estimation period, the estimated factor loading is subjected to errors in variables correlated with the debtto-equity ratio. To see why, let the second risk factor in Equation (1) be the market factor. Let 8

β i2,l denote the true conditional CAPM equity beta, and let ˆβ i2,t denote the conditional CAPM equity beta estimate. Define the estimation error as error i,t ˆβ i2,t β i2,t. Now assume a firm s value increases steadily during the beta estimation period while debt remains relatively constant. Its debt-to-equity ratio drops and its equity beta declines. The equity beta is estimated over a 60-month period, therefore it equals the average of equity betas during these 60 months, which is higher than the true conditional equity beta at the end of the 60-month period. Thus, at the test period, when the debt-to-equity ratio is low, the estimation error is positive (see Figure 2). 15 Similarly, when firm value declines steadily during the beta estimation period, at the test period, the true conditional beta is higher than the beta estimate, and the estimation error is negative. Therefore, the estimation error is negatively correlated with the debt-to-equity ratio of that firm. [Insert Figure 2 here] Now suppose we test a cross-sectional regression model that has the two correct factor loadings, and let the second factor loading be the conditional CAPM beta estimated from the Fama- MacBeth s first-pass regression. In this case, the model tested is well specified, but the CAPM equity beta is estimated with systematic error. Let the model also include either size or book-tomarket denoted by X i,t as one of the independent variables as follows: r i,t = ˆλ 0,t + ˆλ 1,t β i1,t + ˆλ 2,t ˆβi2,t + ˆλ X,t X i,t + ξ i,t. (5) In this case, the time series average of the cross-sectional regression coefficient in the Fama-MacBeth test of the CAPM beta is given as: ˆλ 2 = E t [λ 2,t ] + E t[e i [ β i2,t ]]E t [λ 2,t ]E t [Cov i (ν i,t, error i,t )] Var i ( ˆβ. (6) i2,t ) Since ν i,t is an increasing function of the debt-to-equity ratio, and it has been established that error i,t is negatively correlated with the debt-to-equity ratio, Cov i (ν i,t, error i,t ) < 0. We also have that E t [λ 2,t ] > 0, and E t [E i [ β i2,t ]] > 0, therefore the second term on the right-hand side of Equation (6) is negative, Thus the average slope of the CAPM beta is negatively biased. 15 It is assumed that firm beta and the debt-to-equity ratio are not correlation or that they are not so highly negatively correlated as to negate the results given here. 9

The average slope of size or BE/ME is given as: ˆλ X = E t[(λ 2,t ˆλ 2,t )]E t [E i [β i2,t ]]E t [Cov i (ν i,t, X i,t )] E t [ˆλ 2,t ]E t [Cov i (error i,t, X i,t )]. (7) Var i (X i,t ) To examine ˆλ X, we establish a sign for each term in Equation (7). As the average slope of the CAPM beta is negatively biased, E t [(λ 2,t ˆλ 2,t )] > 0. Second, the term E t [Cov i (ν i,t, X i,t )] has the same sign as the correlation between the debt-to-equity ratio and X i,t. Finally, the term E t [ˆλ 2,t ]E t [Cov i (error i,t, X i,t )] has the same sign as the correlation between the debt-to-equity ratio and X i,t, because the error term is negatively correlated with the debt-to-equity ratio. Therefore, the average slope of X i,t is biased in the direction of the correlation between the debt-to-equity ratio and X i,t. Because size and BE/ME are negatively and positively correlated with the debt-toequity ratio, the capital structure effect predicts that the average slope of size is negative and of BE/ME is positive. The intuition for this result is that the CAPM beta estimated from the Fama-MacBeth firstpass regression entails an error that is negatively related to the debt-to-equity ratio. The effect of this error in variables is to negatively bias the average regression coefficient of the CAPM beta, and to bias the average regression coefficient of other independent variables in the direction of the correlation of that variable and the debt-to-equity ratio. This is the case not only for the CAPM equity beta but also for any factor loading that requires an Fama-MacBeth model first pass regression estimate. 3 Empirical Method There are a two possible ways to test the effect of capital structure. First, test the conditional model 2 E t [r i,t ] = α i,t + ν i,t βil,t λ l,t + λ size size + λ BE/ME BE/ME, (8) l=1 using the Fama-MacBeth test. The null hypothesis is λ size = 0, λ BE/ME = 0, and λ l,t = 0. The problem with this approach is that it is a joint test of three hypotheses: 1) the tested multifactor model is the correct asset pricing model; 2) size and BE/ME are related to expected 10

return due to the capital structure effect; and 3) the market is efficient. If, for example, λ size = 0 or λ BE/ME = 0 is rejected, it may be due to any of these three reasons: 1. the asset pricing model is missing a risk factor and size or BE/ME is a proxy for ν i,t of the missing factor, 2. size or BE/ME is a proxy for a distress risk factor, or 3. small size and large BE/ME firms are mispriced. On the other hand, if we find that λ size = 0 is not rejected, it could be because the CAPM model is not the right model or that the capital structure effect is empirically insignificant. Most cross-sectional tests of asset pricing models that have been proposed report low average R-squares indicating that the models we currently have do not account for all risk factors. 16 Hence, a conditional test will not effectively distinguish the capital structure effect from other explanations. An alternative test method is to compare cross-sectional tests of equity returns and firm returns of the same set of firms. The difference between the time series averages of the regression coefficient of size or BE/ME of firm returns and equity returns measures the capital structure effect separately from other hypotheses outlined above. For example, if size or BE/ME proxy for a distress risk factor, then size or BE/ME should also be related to the cross-section of firm returns. Any priced risk factor should be present in both equity returns and firm returns, and the risk premium should be the same. Moreover, if the stock market is inefficient, the inefficiency should be evident in both equity and firm returns. If size or BE/ME proxy for ν i,t, however, size or BE/ME should be related to equity returns and not firm returns. The objective of this study is to isolate the capital structure effect, so I adopt this method. Test procedure using firm returns has a limitation that the estimated firm returns many still be related to leverage because the estimation procedure does not eliminate the leverage effect from firm returns. For example, the case when the functional relationship between equity and firm returns is firm specific and unaccounted for by a firm return estimation procedure. This may be the case when bankruptcy cost differs greatly among firms, the optimal capital structure differs among firm, and thus the relationship between equity return and firm returns is firm specific. For this cases, a procedure that compares the Fama-MacBeth time series averages of the regression coefficient of equity and firm returns is a conservative test in the sense that it tests for the effect of leverage that is accounted for in constructing firm returns and not for other leverage effects that remain in 16 See for example, Fama and French (1992), Jaganathan and Wang (1996), and Lettau and Ludvingson (2001). 11

firm return estimates. If the difference between time series averages of the regression coefficient of equity and firm returns is significant and estimated firm returns are positively related to leverage, then the overall leverage effects beyond that which is accounted for is even more important. 4 Data The data set consists of all non-financial firms in the intersection of (1) the monthly return files from the Center for Research in Security Prices (CRSP) from July 1963 through June 1998, (2) the merged Compustat annual industrial files from 1962 through 1998, and (3) the CRSP monthly US Government Bills, Notes, and Bonds supplemental files. 17 Consistent with established practice I exclude financial firms because the high leverage associated with them does not indicate financial distress as would leverage in non-financial firms. 4.1 Firm returns Firm returns are obtained from the weighted average of equity returns and bond returns as: r i,t = S i,t V i,t r i,t + D i,t V i,t r B i,t, (9) where S i,t is the value of equity, D i,t is the total value of bonds, and the firm value is V i,t = S i,t +D i,t. Equity returns, r i,t, are from CRSP monthly files. The equity returns and equity value from July of year t through June of year t + 1 are matched with monthly bond returns, ri,t B, and debt value as measured by annual accounting ratios in December of year t 1. 18 Bond returns are returns of Treasury securities of the same maturity as the corporate bond plus a bond risk premium. Maturities of corporate bonds are obtained from Compustat annual industrial files. Compustat reports short-term debt and long-term debt maturing in years 1 through 5 after the year it is reported. Here, debt maturing in one year is computed as the sum of long-term debt in current liabilities (data item 44), taxes payable (data item 71), and notes payable (data 17 I choose the 1962 Compustat start date because of serious selection bias in those data before then (Fama and French (1992)). 18 The implicit assumption here is that, in pricing risk, investors care about annual changes in leverage ratios. 12

item 206). The amount of debt maturing in years 2, 3, 4, and 5 is obtained from data items 91, 92, 93, and 94, respectively. The remaining long-term debt, which equals long-term debt minus long-term debt in current liabilities and debt maturing in less than five years, is assigned an average maturity of ten years. 19 The value of debt of firm i maturing in T years is d i,t+t. The total debt outstanding, D i,t, is the sum of debts maturing in years one through five and debt maturing beyond five years. The total return on debt of a firm is calculated as a value-weighted average of monthly returns on Treasury bonds of matched maturities plus the risk premium p i,t : 5 ri,t B = r g t+t T =1 d i,t+t + r g d i,t+10 t+10 + p i,t, (10) D i,t D i,t where r g t+t is the return on Treasury bonds maturing in T years. The risk premium is obtained using the bond pricing model of Merton (1974). This model is simple, and Sarig and Warga (1989) find that the time profile of the risk premium of corporate bonds is consistent with the theoretical time profile produced by the Merton (1974) model. In calculating the risk premium, I view firm s debt as one composite bond whose time to maturity, τ, is the weighted average of the time to maturity of the bonds outstanding, τ = 5T =1 T d i,t+t D i,t + 10 d i,t+10 D i,t. The risk-free rate to maturity or the yield, y i,t, is the weighted average yield on Treasury bonds, y i,t = 5 T =1 y t+t d i,t+t D i,t d + y i,t+10 t+10 D i,t, where y t+t is the yield of the Treasury bond maturing in T years. The data on Treasury bond yields, bond returns, and the risk-free rate are from the CRSP monthly US Government Bills, Notes, and Bonds supplemental files. 20 The risk premium is given as (Merton (1974)): p i,t = 1 τ ln (1 N ( h i,t ) + V i,t D i,t e y i,tτ N ( h+ i,t )), (11) 19 I extrapolate the graph of the percentage of total debt maturing before year T as a function of T to the time when zero debt remains to obtain this 10 year estimate. 20 One-year, two-year, three-year, four-year, and five-year bond yields are from the Famablis file. The ten-year bond yield is from CRSP fixed-term indices files. The bond returns are from Fama Maturity Portfolio Return files with 12-month intervals. 13

where N ( ) denotes the cumulative normal distribution. The variables h + i,t and h i,t are given by h + i,t ln(v i,t/d i,t ) + (y i,t + δ 2 i,t /2)τ δ i,t τ, and h i,t h+ i,t δ i,t τ. The variable δi,t denotes conditional standard deviation of firm returns. It is obtained by estimating each individual firm s variance using 10-60 monthly approximate firm returns immediately prior to month t. The approximate firm returns are calculated from weighted average of equity returns and bond returns in Equation (10) without the risk premium. Table 2 reports the maturity profile of the debt outstanding, the risk premium, and the bond returns for the entire sample data and for four debt-to-equity ranked portfolios. The fourth portfolio is further sorted on the basis of the debt-to-equity ratio into two portfolios, a and b, of equal number of firms. Portfolio a consists of lower debt-to-equity ratio firms than portfolio b. The average debtto-equity ratio of firms from July 1967 through June 1998 is 3.16. This rather high number is attributable to approximately 1500 data points displaying debt-to-equity ratios in excess of 100. Excluding these data points, the average debt-to-equity ratio is 1.0. I report the results of the test that compares the time series averages of cross-sectional coefficient from Fama-MacBeth tests of equity and firm returns when excluding these 1500 data points in Table A1 in the appendix. The results in Table A1 are the same as the results reported in Tables 5 and 6 below. These high-leverage firms are not driving the empirical evidence presented here. [Insert Table 2 here] The average risk premium is 0.36%, and bond returns are 8.1% per year. Risk premiums range from 0.22% to 0.62%. Bond returns range from 6.86% to 8.91% for firms with low to high debt-toequity ratios. A notable observation is that firms that have a high proportion of current debt are the extreme firms, with either very low or very high debt-to-equity ratios. 21 21 This was also noted by Barclay and Smith (1995). 14

4.2 Other independent variables For the other independent variables in the Fama-MacBeth test, a firm s returns from July of year t through June of year t+1 is matched with accounting ratios as of December of year t 1. BE/ME is the logarithm of the book value of equity (book value of common equity plus balance sheet deferred taxes minus preferred dividends) over market value of equity. The debt-to-equity ratio is denoted by D i,t S i,t, where the book value of debt, D i,t, is the sum of debt maturing in years 1, 2, 3, 4, and 5, and long-term debt maturing beyond five years. S i,t is the market value of equity. The market value of equity (stock price times number of shares outstanding) from the CRSP file for June of year t is used to measure firm size (logarithm of the market value of equity) of a firm s return for July of year t through June of year t + 1. Betas are estimated using the method proposed in Fama and French (1992). A firm s equity beta, β i,t, is obtained by estimating the market model using equity returns and the value-weighted aggregate portfolio of equity returns as the market portfolio. Firm beta, βi,t, is obtained by estimating the market portfolio using firm returns and the value-weighted aggregate portfolio of total firm returns as the market portfolio. For a firm to be included in a test month in the period from July of year t through June of year t + 1, I must have values for total market capitalization for December of year t 1 and June of year t, equity returns and firm returns, equity beta and firm beta, and BE/ME. Since five years of firm returns are needed to compute firm beta prior to the test month, the data set covers 372 test months from July 1967 through June 1998. [Insert Table 3 here] Table 3 reports summary statistics of the data. Average monthly equity returns are 0.88% and firm returns are 0.61%. The average equity beta is 1.11 and the average firm beta is 0.99. While the average equity returns increase from 0.63% to 1.21% for firms with low to high debt-to-equity ratios, the average firm returns remain flat between 0.62% to 0.51 %. 15

4.3 Evidence on size and BE/ME To present an overall picture of how equity returns and firm returns relate to the debt-to-equity ratio after controlling for size or BE/ME, Panels a and b of Table 4 report the average returns of portfolios sorted first on the basis of size or BE/ME and second on the basis of the debt-to-equity ratio. As expected, in all size-sorted or BE/ME sorted portfolios, while average equity returns increase in the debt-to-equity ratio, average firm returns do not. Equity returns also vary with size and BE/ME when portfolios are sorted on the basis of the debt-to-equity ratio. These results suggest that an interaction between size or BE/ME and the debt-to-equity ratio determines average equity returns but not firm returns. [Insert Table 4 here] Table 5 reports the time series average of the monthly cross-sectional regression slope coefficients in the Fama-MacBeth tests that take equity returns as the dependent variable. Because the volatility of equity returns of a levered firm is proportional to ν i,t (Equation (1)), the crosssectional weighted least square regressions are estimated with the residual error terms proportional to ν i,t = (1 + D i,t S i,t )N (h + i,t ) from the Merton (1974) model. The standard errors of the average slope reported are adjusted for any time series correlation in the slope coefficients. The WLS tests are also performed with residual error proportional to the inverse of size (i.e., small firms have high volatility and vise versa). The results are almost identical to those reported in Table 5. [Insert Table 5 here] The Fama-MacBeth regression specifications I through IV in Table 5 confirm evidence elsewhere that size is negatively related to average equity returns and that BE/ME is positively related to average equity returns. The relation of BE/ME is stronger than that of size. The average slope of BE/ME ranges from 0.39% to 0.49% (all significant at the 1% level.) These average slopes are consistent with Fama and French (1992). The average slope of size ranges from -0.09% to -0.14%. The magnitude of the slopes of size is similar to those reported in Fama and French (1992), but size 16

is significant at the 5% level only in regression III when size is the only independent variable. 22 The Fama-MacBeth regression specification V shows that the debt-to-equity ratio is positively related to average equity returns, although its adjusted R-square is small. For illustration, a conditional CAPM beta that is estimated as firm beta multiplied by ν i,t, which is updated every month, is tested along with BE/ME and size as other independent variables in the Fama-MacBeth regression specification VI. The average slope coefficient of the conditional beta is -0.7, which is less negative than the regular CAPM beta, which is estimated using equity returns, in regression specification I. The average slope coefficient of BE/ME in specification VI is 0.40% with standard error of 0.08%. As noted earlier, this result can be interpreted to indicate that BE/ME is a proxy for ν it of a missing factor unrelated to distress or that BE/ME is a proxy for a distress factor. This is why a conditional model does not serve the purpose of isolating the capital structure effect. [Insert Table 6 here] Table 6 reports the time-series average of the monthly cross-sectional regression slope coefficients of the Fama-MacBeth tests with firm returns as the dependent variable. Overall, size and BE/ME are not related to the cross-sectional average of firm returns. In regression I of Table 6, which includes firm beta, size, and BE/ME as independent variables, the average slope of firm beta is 0.11% [0.23%], the average slope of size is -0.06% [0.05%], and the average slope of BE/ME is 0.13[0.07] (standard errors in brackets). None of these values are different from zero at the 5% significance level. In the test that includes size and BE/ME as independent variables, the average slope of size is -0.06% [0.05%] and and BE/ME is 0.11%[0.09%] (regression II). 23 Table 6 also reports the t-statistics (in italics) of the mean difference between time series of cross-sectional regression coefficients of matching independent variables from the Fama-MacBeth tests of equity returns (Table 5) and firm returns (Table 6). For example, the difference between the average slope of BE/ME in the Fama-MacBeth test I of Tables 5 and 6 is -0.26% with a t-statistic of -5.42. Comparison of the Fama-MacBeth tests of equity returns and firm returns for the same set 22 The data set used here is different than the one used in Fama and French (1992) since we require firms to have both equity returns and firm returns. 23 Hecht (2002) finds similar results that BE/ME is substantially less powerful in explaining expected firm returns. 17

of firms shows that the magnitude of the average slope of size declines by 30% to 50%. This drop is significant at between a 5% and 15% level in different regression specifications. The average slope of BE/ME declines by 70%, a decline significant at the 1% level in all regression specifications. The slope coefficient of firm beta in Table 6, although not statistically different from zero, is positive at 0.11. This slope coefficient of firm beta is 0.45% higher than the slope coefficient of the CAPM equity beta reported Table 5 (specification I), and the difference is significant at the 1% level. This is evidence that CAPM betas estimated from Fama-MacBeth first pass regressions using equity returns are negatively biased. Regression specification IV of Table 6 shows that debt-to-equity ratio is not related to average firm returns. The slope coefficient of debt-to-equity ratio is 0.06%[0.09%]. The difference between the slope coefficient of the debt-to-equity ratio in tests of equity and firm returns is statistical significant at a 5% level. 5 Robustness check One drawback in testing firm returns is that bond returns are difficult to obtain and may be noisy. Let the difference between the true firm return and estimated firm return be: err i,t ˆ r i,t r i,t, (12) where a hat over a variable denotes an estimate. For the Fama-MacBeth cross-sectional specification: ˆ r i,t = ˆλ N 0,t + ˆλ l,t X il,t + ξ i,t, (13) l=1 where ξ i,t is random noise and N is the number of independent variables. The estimated average slope coefficient of the Fama-MacBeth test is E t [ˆλ l ] = λ l + E t [Cov i (X il,t, err i,t )/V ar i (X il,t )]. (14) 18

When err i,t is uncorrelated with any other independent variable in the Fama-MacBeth crosssectional regression, the effect of err i,t is to increase the residual error in a cross-sectional regression. But because the time series average of slope coefficients and the t-statistics of this average are calculated using only the point estimate of a cross-sectional regression, err i,t has no effect on these two numbers for large samples. When err i,t is correlated with an independent variable, the bias in the time series average slope coefficient in the Fama-MacBeth test equals the second term in Equation (14). I apply four robustness check. First, I use a bootstrap method to test the hypothesis that the results of the Fama-MacBeth tests of firm returns reported in Table 6 are due to noise. The bootstrap method decouples the relation between equity and bond returns effectively calculating firm returns from equity returns and random noise that has the distribution of the pool of the bond return sample. Firm returns are calculated from weighted average equity and bond returns as r i,t = S i,t V i,t (r i,t + r f ) + D i,t V i,t r B i,t rf. In each Fama-MacBeth test and each test month, the vector of [D i,t, r B i,t ] is randomized and matched with the vector of [S i,t, r i,t ]. Then the Fama-MacBeth test is conducted on the randomized firm returns. I conduct 500 runs for each Fama-MacBeth test and calculate a 5% significance interval for the time series average of each cross-sectional slope coefficient. Table 7 reports these intervals. Slope coefficients reported in Table 6 specifications I and II are outside of the 5% significant interval, thus we can reject the hypothesis that the empirical evidence in Table 6 are due to noise. In a second test, I estimate the extent of bias in the Fama-MacBeth average cross-sectional coefficient by estimating the second term on the right-hand side of Equation (14). The only comprehensive market data on corporate bonds available that I am aware of comes from Lehman Brothers, and even that does not include all bonds from every firm. The bias estimation assumes that the firm returns calculated using quoted prices in the Lehman Brothers bond database are the true firm returns (details on Lehman bond data and firm returns are described in the appendix). err i,t is defined as the difference between Lehman firm returns and firm returns as described in Section 4. The second term on the right-hand side of Equation (14) is estimated by an Fama-MacBeth 19

test of the cross-sectional regression specification: err i,t = b 0,t + N ˆbl,t X il,t + ξ i,t, (15) l=1 where the time series average slope coefficient equals the bias term in the right-hand side of Equation (14): E t [ˆb l ] = E t [Cov i (X il,t, err i,t )/V ar i (X il,t )]. Table 8 reports these bias estimates. The bias in the coefficient of size is positive at 0.09 and statistically significant (regression I). This may cause the difference between firm returns and equity returns reported in Tables 5 and 6. The bias in the coefficient of BE/ME is -0.04 in regression I. This number is small compared to the difference of -0.26 between the coefficients of Fama-MacBeth tests of equity and firm returns reported in Tables 5 and 6. The bias in the coefficient of BE/ME is -0.12 in regression III, again small compared to the difference of -0.33 between the coefficients of Fama-MacBeth test of equity and firm returns reported in Tables 5 and 6. It is possible that errors in firm returns account for the results on size reported in Table 6. But errors in firm returns do not account for the result reported in Table 6 on BE/ME. The third robustness check compares the Fama-MacBeth tests of equity returns and firm returns constructed using the Lehman bond data. The Lehman firm returns are used only for a robustness check because the data set is very small. 24 Panel a of Table 9 reports the sample statistics. Panel b of Table 9 reports the time series average of cross-sectional coefficients of the Fama-MacBeth tests of equity returns and firm returns for the same set of firms. BE/ME is related to the cross-section of equity returns but not of firm returns. The t-statistics (in italics) indicate the mean difference between the average cross-sectional regression coefficients of matching independent variables in the Fama-MacBeth tests of equity and firm returns. The difference between the average slope coefficients of BE/ME in Fama-MacBeth test of equity and firm returns in regression II is -0.14 (significant at a 5% level). In this data set, size is not related to average equity returns. 24 Calculating firm returns using the Lehman database poses drawbacks due to the limited data available. The acquisition of accurate historical bond price data is a very difficult task; see Warga and Welch (1993). The data set available is small and the data rarely include the market value of all the debt outstanding of a firm. Matching firms in CRSP and in the Lehman bond files from 1973 through 1997, we can obtain bond returns (from quoted prices) for an average of only 250 firms each month, and for many of these firms, these bonds account for less than one-third of all debt outstanding. If bond returns from matrix prices are included, we obtain an average of 350 firms each month. This small data set significantly reduces the power of the test (compared to the method described in Section 4.) 20

The fourth robustness check uses a different method to construct firm returns. Ideally, if we know the relation between equity price and firm value we can calculate a firm s excess returns by dividing the excess equity return in Equation (1) by ν i,t. A firm s excess return is given by r i,t = r i,t ν i,t = 2 l=1 β il,t λ l,t + 2 l=1 σ il,t dw l,t + σ i,t dɛ i,t. (16) ν i,t from the Merton (1974) model is employed to obtain the unlevered firm returns. The relationship between average firm returns and size and BE/ME is tested. The test results, reported in Table A2 in the appendix, are qualitatively the same as the results in Section 4. Overall, empirical evidence from the these robustness checks continue to support the conclusion that large portions of the relationship of equity returns and size or BE/ME are due to the capital structure effect. Although the support is weaker for size. 6 Concluding remarks This study examines why average equity returns are related to size and BE/ME. Answering the question is central to understanding what determines expected stock returns. The evidence uncovered here suggests a new explanation. In a multifactor model of financially levered firm s expected equity returns, the factor loading equals the firm-level factor loading multiplied by the price elasticity of equity value with respect to firm value, ν i,t, which is time varying. This elasticity is a non-linear function of the firm s leverage measured in market value terms. Therefore, change in equity values leads to changes in this price elasticity multiplier. This study examines the hypothesis that size and BE/ME are related to equity returns because they proxy for the price elasticity effect present in equity-level factor loadings. Here the effects of changing capital structure on equity returns is termed the capital structure effect. The theoretical analysis of the capital structure effect yields very different empirical implications and interpretations than is common in the asset pricing literature. First, size and BE/ME will be related to expected equity returns when the asset pricing model tested is missing any risk factor. This is because size and BE/ME proxy for price elasticity of the missing factor. Second, even if the 21