Leverage, beta estimation, and the size effect

Transcription

1 Leverage, beta estimation, and the size effect Wolfgang Drobetz Iwan Meier Jörg Seidel HFRC Working Paper Series No.1 5. September 2014 Hamburg Financial Research Center e.v. c/o Universität Hamburg Von-Melle-Park Hamburg Tel (0) Fax: 0049 (0) info@hhfrc.de

2 Leverage, beta estimation, and the size effect * Wolfgang Drobetz Iwan Meier Jörg Seidel S September 5, 2014 A measurement error in beta that arises from changes in leverage during the beta estimation window contributes in explaining the size effect. Simulations of asset returns show that the magnitude of the bias in equity returns is proportional to the stock market-induced changes in leverage. We propose a point-in-time beta that incorporates leverage at the end of the beta estimation window rather than the average leverage during this window. Using the point-in-time beta to compute expected stock returns for historical U.S. data, we document that the size effect is substantially reduced. In contrast to other explanations of the size effect, our approach does not introduce market frictions or additional risk factors. * We thank John Doukas, Michael Halling, Tatjana Puhan as well as participants at the 2010 Financial Management Association (FMA) Meeting in New York and the 2011 European Financial Management Association (EFMA) Meeting in Braga for helpful comments. Wolfgang Drobetz, School of Business, University of Hamburg, Von-Melle-Park 5, Hamburg, Germany. Mail: wolfgang.drobetz@wiso.uni-hamburg.de. Iwan Meier, HEC Montréal, 3000 Chemin de la Côte-Sainte-Catherine, Montréal (Québec), Canada, H3T 2A7. Mail: iwan.meier@hec.ca. S Jörg Seidel, School of Business, University of Hamburg, Von-Melle-Park 5, Hamburg, Germany. Mail: joerg.seidel@wiso.uni-hamburg.de.

3 1 Introduction The size effect (Banz, 1981; Reinganum, 1981) has been studied extensively in the asset pricing literature. The observation that small stocks outperform large stocks contradicts the pricing restriction that expected excess returns are fully explained by their covariance with the returns on the market portfolio and poses an anomaly of the Capital Asset Pricing Model (CAPM). Fama and French (1992, 1993) note that neglecting the size effect, while relying on the model s prediction that differences in betas explain the cross-section of expected returns, may lead to errors in asset pricing tests. As the true underlying beta is unobservable, empirical tests of the CAPM estimate single security or portfolio betas. These tests assume that the estimated betas are unbiased predictors of the true betas. Several potential biases in the estimated betas have been described in the prior literature, ranging from data deficiencies (such as thin or asynchronous trading and survivorship biases) to stochastic effects. However, little research has focused on how changes in financial leverage distort beta estimates. The aim of our study is to close this gap in the literature and show how the bias from leverage changes affects expected returns and induces a size effect. Combining proposition II of Modigliani and Miller (1958) with the CAPM, Hamada (1972) shows how the equity beta reflects financial leverage. In fact, any change in market leverage implies a linear change in the equity beta and in expected stock returns. In empirical asset pricing tests, betas must be estimated over longer time windows of up to five years to reduce estimation errors. These estimated betas incorporate a firm s average debt-to-equity ratio during the estimation window rather than its leverage at the end of the estimation window. Since returns subsequent to the beta estimation window are expected to be linear in market leverage at the end of the estimation window, expected returns need not be linear in estimated betas based on the average debt-to-equity ratios. Or put differently, the expected returns need not be linear in the estimated betas. Using simulated returns for size-sorted portfolios, we show that the estimated beta is a biased proxy for the true point-in-time beta at the end of the estimation window. In particular, increasing leverage during the estimation window will lead to a downward bias in the estimated beta. This measurement error in betas introduces a bias in expected stock returns, and the bias is most pronounced for small firms. An example highlights the intuition of our analysis. Assume an asset value of A 0 = 10, a debt value of D 0 = 4, and an initial debt-to-equity ratio of L 0 = D 0 /E 0 = 4/6 = Moreover, assume an asset beta of β A = 0.5 and an equity risk premium of 2

4 8%, implying an expected excess asset return of E[R A ] = 4%. Figure 1 illustrates two possible simulation paths, where the asset value either increases to A t = 20 or decreases to A t = 5 during the following 60 months. For the upper simulation path (past winner and large capitalization firm 1), we have: E t = 16 and L t = D t /E t = 4/16 = Since the average equity value during the estimation window is E 60 = 11, for large firms the average leverage during the beta estimation window ( L 60 = 0.40) is higher than the leverage at the end of the beta estimation window (L t = 0.25). 1 Following Hamada (1972), we have β t = β A (1+L t ) = 0.5 (1+0.25) = 0.625, implying an expected excess equity return of E[R E,t ] = 5%. The average beta during the beta estimation period is β 60 = β A (1+ L 60 ) = 0.5 (1+0.4) = 0.7, resulting in E[R E,60 ] = 5.6%. Accordingly, the (expected) pricing error for large stocks is relatively small with P E 60 = 0.6%. In sharp contrast, for the lower simulation path (past loser and small capitalization firm 2), we get: E t = 1, L t = D t /E t = 4/1 = 4, E 60 = 3.5, and L 60 = 1.5. Therefore, for small firms the average leverage during the beta estimation period ( L 60 ) is smaller than the leverage at the end of the beta estimation period (L t ). Furthermore, we have β t = β A (1+L t ) = 0.5 (1+4) = 2.5 with E[R E,t ] = 20% and β 60 = β A (1+ L 60 ) = 0.5 ( ) = 1.25 with E[R E,60 ] = 10%. Accordingly, the (expected) pricing error for small stocks is large with P E 60 = +10%. The pricing error changes its sign from negative to positive, thus the CAPM underestimates the returns of small stocks. The pricing error is asymmetric because the debt-to-equity ratio is a convex function of the value of equity. Therefore, a change in equity has a much stronger effect on the debtto-equity ratio for a high leverage firm than for a low leverage firm. [Insert Figure 1 here] Market leverage changes with stock returns (Welch, 2004). Firms with negative (positive) stock returns over any given period of time will exhibit, on average, a lower (higher) market value of equity at the end of this period than within this period. Therefore, the estimated equity betas of firms with negative past returns will be biased downward compared with the true point-in-time betas at the end of the measurement window. Any test for linearity between estimated betas and subsequent stock returns will lead to a measurement error and higher-than-expected returns for small firms. The pricing error is smaller for large firms because the debt-to-equity ratio is a convex function of the value of equity. This mechanism is able, at least partially, to explain the size effect. 1 These averages must be computed from the simulation path due to path-dependency. 3

5 We propose a simple point-in-time correction for the bias in beta estimation. Adjusting the beta for changes in financial leverage during the beta estimation window, we are able to substantially reduce the size effect for both simulated and historical stock returns. The abnormal returns of small firms, many of which have been losers during the beta estimation window, become statistically insignificant in both cases. Our study contributes to the literature in three ways. First, using a simulation setup with minimal assumptions, we show that the estimated betas suffer from a bias which is correlated with past stock returns. In particular, the common methodology for estimating betas creates a bias in expected stock returns for past losers. Second, we demonstrate that the size effect is attributable to this measurement error in betas. The bias in betas induces a size effect of an order of magnitude similar to historical data. This result holds both for pre-ranking and post-ranking betas. Finally, we suggest a simple correction for the bias and show that a point-in-time beta is able to substantially reduce the size effect. Taken together, the size effect that is observed in stock market data should not simply be considered to be a CAPM anomaly, but rather, at least partially, to be an artifact of the beta estimation methodology. The remainder of this study is organized as follows. Section 2 reviews the theoretical foundations. Section 3 describes our simulation framework, and section 4 presents our testable hypotheses. Section 5 summarizes the simulation results, while section 6 applies our suggested correction for the point-in-time beta on U.S. stock market data. Finally, section 7 concludes and provides an outlook for future research. 2 Theoretical foundations and simulation setup 2.1 Changing leverage and point-in-time betas Starting from Modigliani and Miller s (1958) proposition II, Hamada (1972) derives a relationship between a firm s leverage and its equity beta. Assuming risk-free debt, the equity beta, denoted as β E, is the levered version of the firm s asset beta, labeled β A : β E = β A ( 1 + D ), (1) E where D is the debt-to-equity ratio. Galai and Masulis (1976) develop an option E pricing model to further investigate the capital structure effects on a stock s systematic risk. They argue that a stock s systematic risk is the product of the firm s systematic 4

6 risk and the price elasticity of the equity value with respect to changes in the firm s asset value. Based on Merton (1974), the equity beta is not only a positive function of leverage, but also a negative function of the risk-free interest rate r f, the variance of the asset returns σ 2, and the time to maturity T. Relaxing the assumption of risk-free debt, Galai and Masulis (1976) derive the following generalized version of equation (1): β E = β A N(d 1 ) d 1 ln(v/c) + (r f σ 2 )T σ T ( 1 + D ), (2) E. (3) where V and C denote the value of the firm and the face value of debt, respectively. DeJong and Collins (1985) use this framework to identify the sources of beta instability. In line with equation (2), they document that firms with higher leverage exhibit greater beta instability, and they conclude that the higher residuals for high leverage firms in market model regressions are attributable to capital structure effects. However, DeJong and Collins (1985) do not investigate the unbiasedness of the estimated betas. Hecht (2002) and Choi (2009) use stock and bond returns to analyze the cross-section of firms asset returns. Compared to stock returns, they show that the patterns of bookto-market, reversal, and momentum effects are less pronounced or even non-existent in asset returns. Therefore, capital structure effects play a major role in generating these stock return properties. Charoenrook (2004) documents that Fama and MacBeth (1973) estimates are systematically biased in the presence of leverage. The bias is caused by the assumption that factor loadings are constant during the estimation period, which leads to a systematic relationship between expected stock returns and size (and bookto-market). Our study adds to this strand of research by investigating the magnitude of the bias in estimating equity betas that is caused by market-induced changes in leverage. We simulate asset returns with risk-free debt, as assumed in Hamada (1972). We also assume risk-free debt because simulating risky debt would require a pricing model for debt, and any mispricing of debt would cause a mispricing of equity. In fact, we choose a simulation approach that creates a clean-room sample and excludes many other explanations for the size effect, especially those related to data deficiencies. For example, Roll (1981) argues that thin-trading is responsible for the size effect. While thin-trading is an observable effect in market data, in our simulation setup no return is subject to thin or asynchronous trading. We exclude any survivorship bias in simulated returns 5

7 (Kothari et al., 1995), and we use the true market portfolio in the beta estimation (Stambaugh, 1982; Roll and Ross, 1994; Ferguson and Shockley, 2003) to avoid possible mean-variance inefficiencies of the market portfolio proxy. Furthermore, our findings are unaffected by data snooping (Black, 1993). No additional risk factors can influence our results. For example, Amihud and Mendelson (1989) attribute the size effect to a liquidity premium, and the findings in Chan et al. (1985) point to the influence of a distress factor. Keim (1983) documents a relationship between the size effect and the January effect. Klein and Bawa (1977) analyze the problem of investors when the distribution of securities returns has unknown parameters. In this case, estimation risk induces the average investor to hold a portfolio that differs from the mean-variance optimum. Kraus and Litzenberger (1976) examine investors preference for higher moments of the return distribution. Finally, Berk (1995) argues that any missing risk factor will be proxied by firm size due to discounting effects. All these factors are excluded from our simulation framework and cannot impact our results. We infer equity returns of leveraged firms from simulated asset returns with constant asset betas. In other words, we simulate equity returns for partly debt financed firms, assuming that the CAPM holds unconditionally for firms asset returns. In this simplistic simulation setup, we observe a size effect that is caused endogenously by changes in leverage during the beta estimation window. Moreover, we propose a simple correction for the bias in the estimated equity beta. The equity beta estimated over the preceding 60-month time window, denoted as β 60, incorporates the average debt-to-equity ratio during the estimation window, L 60, rather than the leverage at the end of the estimation window, L t. Based on equation (1), we derive a firm s asset beta, denoted as β A, from dividing β 60 by 1 + L 60 for each firm. Assuming that the asset beta does not change during the estimation window, we use the leverage at the end of the estimation period to compute a point-in-time equity beta, denoted as β t : β t = β A (1 + L t ) = β 60 (1 + L t) (1 + L 60 ). (4) Correcting the estimated equity beta for leverage changes during the estimation window removes the size effect in simulated data and reduces it substantially using market data. However, a caveat is in order for equations (1) and (2), as they are only correct in continuous time. Using these equations to compute point-in-time betas and to forecast discrete returns may also lead to biased results. The two equations implicitly assume that the equity beta remains constant; they ignore that the return over the subsequent 6

8 period may affect leverage and the equity beta in discrete time. Our simulation results indicate that this inaccuracy does not lead to substantial biases as long as the return measurement period is short enough and the initial leverage is not excessively high. However, if leverage is high, equation (4) delivers upward-biased point-in-time betas even with short time intervals. If a highly leveraged firm does not default, its equity value is likely to increase sharply, the level of leverage decreases, and thus the point-intime estimate will be overestimated. On the one hand, this effect can be mitigated by computing the point-in-time beta whenever a stock price is observable. On the other hand, by using very short time intervals one may encounter other issues, such as poor return estimates due to thin trading. 2.2 Post-ranking betas Already Chan (1988) discusses time-variation in the market risk premium and beta measurement errors as sources of abnormal returns. He finds that the loser portfolio, on average, loses 45% in market value during the three years before portfolio formation, while winners gain 365% during the same time window, implying that the leverage and the risk of loser (winner) stocks has increased (decreased). Since estimating betas before sorting stocks into size portfolios (pre-ranking betas) generates a downward bias in betas for past losers, Chan (1988) proposes estimating betas after portfolio formation (postranking betas). Following Fama and French (1992), most asset pricing studies have used post-ranking betas as opposed to pre-ranking betas. However, using post-formation betas will likely also not lead to unbiased estimates. Figure 2 illustrates that the bias stems from leverage changes in the portfolios during the beta estimation period, even if this is a period after portfolio formation. A highly leveraged firm is likely to have extreme (very high or very low) equity returns. On the one hand, if returns are very high, portfolio leverage sharply decreases compared to the portfolio formation period. On the other hand, if returns are very low, a firm has a high chance to default and thus to be removed from the sample. The overall effect is similar in both cases because the average leverage of the remaining firms in the portfolio is lower compared to the portfolio formation period. [Insert Figure 2 here] Holding asset volatility constant, high asset returns imply low default rates, but sharp leverage decreases due to high expected equity returns. In contrast, for a given level of 7

9 asset volatility, low asset returns lead to higher default rates, thus systematically removing highly leveraged firms from the sample during the (post-ranking) beta estimation window. Depending on the relationship between expected assets return and volatility, one or the other effect will dominate. Since both high and low asset returns lead to a decrease in average portfolio leverage, however, both cases will deliver downward-biased beta estimates. Therefore, we also expect a size effect for post-ranking betas. Our simulation results confirm this conjecture and show that the overall beta bias is of the same order of magnitude as the bias for pre-ranking betas. We conclude that simply shifting the beta estimation period is not a solution to effectively mitigate the size effect. 2.3 The simulation setup In our empirical framework, we simulate the market value of assets, denoted as A i,t, for all firms i (i = 1,..., N) with a given amount of debt, denoted as D i,t. The market value of equity, labeled E i,t, is a residual claim: E i,t = A i,t D i,t. (5) From these simulated equity values we compute equity returns and examine whether a size effect shows up in the simulated data. The choice of the processes for changes in A i,t and D i,t is essential. The parameters must be chosen such that no size dependency is induced and that the Capital Asset Pricing Model (CAPM) holds for asset returns. As E i,t and D i,t are linearly dependent, any size dependency in the process for the market value of debt will spill over to the simulated equity values. For example, if debt returns increase with asset size, equity returns would be higher for smaller firms. However, this size effect would be artificially caused by the process for the return on debt. In order to avoid any biases that result from the pricing model for debt, we choose the simplest possible setup with risk-free debt of zero maturity. Accordingly, the market value of debt is equal to its face value at any time. Risk-free debt implies that a firm can be liquidated at no cost. If a firm s asset value falls and becomes equal to the face value of debt, the firm will be liquidated and its debt will be fully repaid immediately. With debt having zero maturity, its value is also unaffected by changes in the risk-free interest rate. Assuming that debt is risk-free and has zero maturity, the market value of debt equals its face value at any point in time (i.e., debt is always priced at par). Risky debt or longer maturities require an explicit valuation model for debt, and any model error potentially causes artificial effects in equity returns. Empirical tests become joint 8

10 hypothesis tests of the patterns in equity returns and the valuation model for debt. For further simplification, we assume that the risk-free interest rate is constant and zero. Asset returns follow a geometric Brownian motion with a continuously compounded expected rate of return and a continuously compounded variance. 2 We assume that the CAPM holds unconditionally for asset returns. There exists an unobserved market portfolio, and expected asset returns are proportional to their covariance with the returns on the market portfolio. By construction, the covariance with the market portfolio is the only source of expected returns. With a zero risk-free interest rate, the expected asset return for zero-beta firms is equal to zero. Assuming that there is idiosyncratic volatility in asset returns, the correlation between returns on the market portfolio and asset returns is imperfect. Therefore, the instantaneous continuously compounded asset return for firm i at time t, denoted as ˆr A,t,i, is given as: ˆr A,t,i = γ A,i (μ M + σ M ε M,t ) Systematic return + σ A ε i,t 0.5 σa 2. (6) Idiosyncratic return The return generating process in equation (6) is driven by normally distributed market portfolio returns, denoted as μ M + σ M ε M,t, and normally distributed firm-specific (idiosyncratic) returns, labeled σ A ε i,t. μ M is the expected return of the market portfolio, and ε M,t are independent standard normal random numbers that capture innovations in the market portfolio. The coefficient γ A,i captures the linear dependency of asset returns on market returns similar to an asset beta. The random variable ε i,t is firm specific and drawn from an independent standard normal, and the term 0.5 σa 2 ensures that the expected discrete idiosyncratic return equals zero. In line with the Capital Asset Pricing Model, the covariance with the market portfolio is the only source of expected asset returns. We simulate instantaneous continuously compounded asset returns. 3 ˆr A,t,i is normally distributed due to its linear dependency on ε i,t and ε M,t. The parameter γ A,i is set equal to one for all firms, ensuring that differences in the cross-section of equity returns are caused by differences in leverage and not by differences in asset betas. 2 Although asset values are not directly observable, many standard models are based on this assumption (Merton, 1974). We conjecture that financial leverage changes the moments of equity returns. The same effect is likely to be the case for asset returns in the presence of operating leverage, but our analysis is restricted to financial leverage. 3 Discrete returns cannot be normally distributed as assumed by the CAPM because the largest loss an investor can realize is 100%. In contrast, the normal distribution supports the entire real line. 9

11 Asset values are simulated over some period of time. Therefore, we need to define a continuously compounded asset return, labeled r A,[t,t+1],i, which translates ˆr A,t,i into a return in finite time: ˆr A,t+1,i, if t [t, t + 1] : ˆr A,t,i > ln(d) ln(a) r A,[t,t+1],i = (7) ln(d) ln(a) otherwise. According to equation (7), r A,[t,t+1],i is equal to ˆr A,t+1,i if the asset value is greater than the value of debt during the entire time interval. However, if at any point during this time interval the firm s asset value drops to the value of debt, thus if the asset return is ln(d) ln(a), then r A,[t,t+1],i will also be equal to ln(d) ln(a). Equation (7) further ensures that the continuously compounded asset return during an arbitrary time interval from t to t + 1, r A,[t,t+1],i is in line with our assumption of risk-free debt. If the check for liquidation was only made at the end of each period, asset values could potentially fall below the face value of debt. Any losses incurred by the lender would contradict our assumption of risk-free debt. Another important property of our setup is that equation (7) does not add a trend to asset returns. If ˆr A,t+1,i has an expected value of zero, then r A,[t,t+1],i also exhibits an expected rate of return of zero independent of the choice of leverage. 4 The assumption of zero maturity debt is convenient from a mathematical point of view, and it seems to be realistic from an economic standpoint. Firms tend to use current accounts and liquidity lines for their cash management. 5 As soon as their asset value drops below the notional amount of total debt, their current account and any other access to the money market will dry up, and firms will default on lack of liquidity. This may even be the case in the presence of other tranches of debt with longer maturities. The timing of default is likely to be triggered by debt with the shortest time to maturity, and in many instances this is daily revolving debt. However, our approach in equation (7) comes at a cost. Specifically, it implies that the distribution of asset returns must account for path dependency, as illustrated in Figure 3. In order to estimate the probability density of an asset value at the end of the period, A i,t+1, we need to integrate over all possible paths leading to this point from 4 In contrast, with risky debt high equity returns result if risk-shifting activities become a problem. While risk-shifting causes excessive equity returns if the market value of debt drops below the face value and asset values are being redistributed from debtholders to shareholders, our simulation results show that supposedly abnormal returns even occur in the absence of risk-shifting. 5 According to Barclay and Smith (1995), firms with high leverage have a particularly high proportion of current debt. 10

12 the starting value, A i,t. To this end, we employ the path integral by Feynman (1948). Because no analytical solution to the path integral in equation (7) exists, we solve the problem numerically by slicing each period into 1, 000 time slices and check for default at the end of each time slice. The high number of slices ensures that the pricing kernel is approximated with sufficient accuracy and that any remaining losses to debtholders are negligible compared to the observed effects in equity returns. [Insert Figure 3 here] We use the continuously compounded return from equation (7), r A,[t,t+1],i, to calculate the change in asset values from time t to t + 1 as follows: A i,t+1 = A i,t e r A,[t,t+1],i. (8) As the expected asset return is positive, firms will either be liquidated if they hit the debt barrier, or they will grow large and the debt-to-equity ratio decreases. Therefore, we need a rule to keep leverage in a plausible range and force our simulation to converge to a distribution with realistic debt ratios. Specifically, we assume that each simulated firm pays a dividend if its debt-to-equity ratio, L i,t, falls below This cash dividend subsequent to an increase in equity values will be financed by the issuance of new debt, and the amount to be issued and distributed guarantees that leverage is 0.5 after the dividend has been paid. The trigger level is chosen such that the average leverage ratio for the largest size decile will approximately match the empirical leverage ratio in the largest size decile of U.S. firms in the Compustat Global database, which is In order to adjust the debt-to-equity ratio to a minimum leverage of 0.5, the dividend, denoted as d i,t, must be: ( d i,t = max 0, A ) i,t 3 D i,t. (9) The max( ) function ensures that dividends are only paid if leverage falls below the lower limit. 7 The choice of the dividend payment does not strongly influence the results 6 This model is simplistic, but it suffices for our purpose. In practice, additional factors will influence the dividend payments, such as information asymmetry and agency problems (Allen and Michaely, 2003). 7 To see that this simple rule implements the desired target leverage, assume that D = 2, E = 10, and thus A = 12. According to equation (9), the dividend is A 3 D = 4 2 = 2. After the dividend has been paid out, D = = 4, E = 10 2 = 8, and L = 4 8 =

13 of our simulation because our primary interest is on high leverage firms, which are mainly small firms with negative prior returns. These firms never hit the lower trigger for leverage, and thus do not distribute dividends. For debt-to-equity ratios above 0.5, all changes in leverage are induced by equity returns. Any payment of dividends occurs at the end of the period [t, t+1] during which the trigger level has been reached. Dividends are paid to shareholders and increase their total equity returns, denoted as R E,[t,t+1],i. Dividends are financed by debt issuances, and thus the level of debt is increased by the same amount. Taken together, we have: R E,[t,t+1],i = (E t+1,i + d i,t+1 E t,i )/E t,i, (10) D i,t+1 = D i,t + d i,t+1, (11) L i,t = D i,t E i,t. (12) The novel idea of our simulation analysis is to demonstrate that there is an endogenous relationship between the market value of equity and the firm s leverage. To make sure that this size effect is generated endogenously, we choose the starting values for the simulation such that there is no relationship between a firm s market value of equity and its leverage at the beginning of the simulation: E i,0 = ν 10, (13) L i,0 = ν 2, (14) D i,0 = E i,0 L i,0, (15) A i,0 = D i + E i,0. (16) To generate dispersion in leverage ratios, ν denotes equally distributed random numbers in the interval from zero to one. By drawing the random variates for the starting values of equity, E i,0, independently from the starting values of leverage, L i,0, we ensure that there is no correlation between these two firm characteristics at the beginning of the simulation, thus any observed size effect must be endogenous. In addition, by choosing the values for equity and leverage, we also determine the starting values for debt, D i,0, and total assets, A i,0. Finally, we construct the return on the market portfolio, labelled R M,t, as the crosssectional value-weighted average of all simulated equity returns. 8 The return on the 8 Ferguson and Shockley (2003) argue that using an equity-only proxy for the market portfolio may cause a size bias in the estimated equity betas if the firms equity returns covary with the omitted 12

14 market portfolio is given as: R M,t = N i=1 R E,t,i E t 1,i N i=1 E. (17) t 1,i Overall, the framework described in this section allows us simulating a large set of firms with their specific asset returns, debt-to-equity ratios, and equity returns. Therefore, it enables us to examine whether the size effect shows up in this very general distribution of equity returns. 3 Testable hypotheses We use simulated equity values and returns to examine whether the size effect shows up in our clean-room sample. Following Reinganum (1981), in each month t we sort firms into size deciles according to their simulated market value of equity, E i,t. Firms with an equity value of zero in period t are marked as defaulted; they are not further considered in this and all later periods. However, in period t they are assigned a return R E,t,i = 100% to avoid a survivorship bias. We form an equally-weighted portfolio for each size decile. Using portfolio returns over the period from t 60 to t, we estimate pre-ranking portfolio betas, β 60, from regressions on the simulated market returns, R M,t. We also record the subsequent portfolio returns over the period from t to t + 1. Portfolios are formed in each simulation month t, and both the average return and the average beta are computed for each decile over time. As there is no size specific factor in our simulations, we expect that each size portfolio has the same average beta and exhibits the same average return. Accordingly, our first two testable hypotheses are: Hypothesis H1: The average estimated betas of all size portfolios are equal. Hypothesis H2: The average estimated returns of all size portfolios are equal. If hypotheses H1 and H2 do not hold, and given that the CAPM holds for asset returns by construction, we expect that any cross-sectional variation in returns can be explained by variation in betas. This leads to an alternative testable hypothesis: assets from the market portfolio. However, this problem should not be important in our simulation setup because equities are the only return-generating securities. 13

15 Hypothesis H2.a: There is a linear relationship between the average estimated portfolio returns and the average estimated betas. Equation (14) ensures that leverage is independent of the market value of equity across firms at the beginning of our simulations. The market value of equity changes, thus the debt-to-equity ratio also changes along each simulation path. Therefore, we test if the average leverage is equal across all size deciles: Hypothesis H3: The average leverage is equal across all size deciles. As Modigliani and Miller (1958) point out, one expects higher equity returns for firms with higher financial leverage. If hypothesis H2.a does not hold, we expect that changes in leverage (at least partially) explain the deviation from a linear relationship between the average estimated returns and the average estimated betas. 4 Simulation results 4.1 Leverage and pricing errors We run our simulations for N = 80, 000 firms, and each firm s asset returns are simulated over T = 500 months. Furthermore, we assume an idiosyncratic monthly asset return volatility of σ A = 0.05, a monthly volatility of the market portfolio of σ M = 0.005, and a monthly expected return of the market portfolio of μ M = The numbers are chosen to generate equity returns that are comparable to those in the historical sample of U.S. firms (see section 6). However, our main simulation results are not limited to this particular set of parameters, and findings with different sets of parameters are qualitatively similar. Table 1 presents the first five moments of all simulated return series. The distribution of continuously compounded asset returns is close to a normal distribution, despite the asymmetry introduced in equation (7). The number of firm liquidations is too small to have a pronounced effect on the overall return distribution. Nevertheless, the deviation in skewness (0.149) is caused by the path dependency in equation (7) and disappears if the simulation omits debt. [Insert Table 1 here] The effects are different for continuously compounded equity returns. They exhibit an increased kurtosis of (instead of 3, the value for a normal distribution). The 14

16 signs of the uneven moments are negative in the presence of leverage, while they are positive for asset returns. Equation (7) introduces a positive skewness to asset returns; they can grow unlimited but have a floor at ln(d) ln(a). In contrast, continuously compounded equity returns have no floor; an asset return of ln(d) ln(a) implies an equity return of. While the probability of an ever increasing return converges to zero, the probability of a return of is larger than zero and finite. Although we remove all infinite returns from the sample when computing moments, the remaining effect is strong enough to reverse the positive skewness in asset returns. Without debt all returns are normally distributed, thus the skewness in returns is a function of leverage. Transformed into discrete returns, the skewness (0.476) and the fifth moment (49.835) of equity returns are positive because the loss is limited. To test our hypotheses developed in Section 3, we sort firms each month into decile portfolios according to their size, as measured by the market value of equity. For each decile portfolio and each month we estimate the pre-ranking beta, β 60, using the market returns over the 60 months prior to the formation month. The results are shown in Table 2 and visualized in Figure 4. Each decile portfolio consists of about 5,335 firms, on average. The estimated beta, β 60, ranges from for the firms with the highest market value of equity (decile 10) to for the firms with the lowest market value of equity (decile 1). This difference in betas is statistically significant (based on a t-test for differences in means), thus we reject hypothesis H1 that the estimated betas are equal across all deciles. Although we exclude any size dependency in our simulations, there is a clear relationship between firm size and estimated betas. Moreover, we reject hypothesis H2 that returns are equal across all decile portfolios. Monthly excess returns range from 0.811% for large cap stocks to 1.466% for small cap stocks; the difference is statistically significant. [Insert Table 2 and Figure 4 here] In a next step, we compute expected excess returns, denoted as E[R E ], by multiplying the estimated betas, β 60, with the average excess return on the market portfolio, E[R M ]. The simulation results in Table 2 reveal that the relationship between the average realized (simulated) excess return, R E, and the average expected excess return, E[R E ] = β 60 E[R M ], is non-linear. The pricing error, defined as R E E[R E ] and labeled P E(β 60 ), is relatively small for large cap stocks; it is smaller than 0.06% per month for each of the nine largest size portfolios (deciles 2 to 10). In contrast, the bottom decile portfolios exhibit a monthly pricing error of 0.308%; it is statistically larger than zero, 15

17 with a t-value indicating significance at the 1% level. Overall, our simulation produces a relationship between size, beta, and stock returns, which is in line with market data. The returns of small capitalization stocks are too high to be explained by differences in betas, and we observe a CAPM anomaly in the smallest size decile. In contrast, the pricing errors for large capitalization firms are negligible. Table 2 further reveals that in the bottom half of the size decile portfolios with small capitalization stocks the average market capitalization at the end of the beta estimation window, ØE t, is smaller than the average market capitalization within this time window, ØE 60. For the top half of the size decile portfolios containing large capitalization stocks, the relationship is the other way round; on average, the size at the end of the estimation window is larger than within. This observation indicates that sorting firms by market capitalization implicitly sorts firms by past returns in our framework. Small firms have a higher probability of having suffered from prior negative returns, and vice versa. We assume that firms pay a dividend if the debt-to-equity ratio falls below 0.5. Firms with positive returns and lower leverage counterbalance the market-induced changes in leverage by paying a higher dividend. In contrast, small capitalization stocks are unable to react to changes in market leverage. Consistent with the empirical evidence for stock price run-ups prior to equity issuances (Baker and Wurgler, 2002; Chang and Dasgupta, 2009), our setup does not allow firms to raise equity to compensate past losses. This effect results in increased leverage for past losers and in a correlation between leverage and size, thereby violating hypothesis H3. In fact, Table 2 reveals higher debt-to-equity ratios for small capitalization stocks than for large capitalization stocks. Both the average leverage during the estimation window, denoted as ØL 60, and the average leverage at the end of the estimation window, labeled ØL t, are decreasing with increasing size. Our simulation framework does not impose any correlation; by construction, at the start of our simulations the average leverage is equal across all size deciles. Small capitalization stocks tend to suffer from a negative performance during the beta estimation window. Their average market capitalization at the end of the estimation window, ØE t, is smaller than both the average market capitalization at the beginning of the estimation window, ØE 0, and the average market capitalization within the estimation window, ØE 60. A related implication is that the average debt-to-equity ratio at the end of the estimation window, ØL t, is higher than the average leverage during the estimation window, ØL 60 for these firms. While β 60 is estimated over the prior 60 months and reflects the risk of the average leverage during the estimation window (preranking beta), the expected return in the following month is proportional to a firm s 16

18 leverage at the end of the estimation window. As long as the average leverage is a good proxy for the period-end leverage, this is not a problem. However, Table 2 indicates that this approximation is inappropriate for the small size decile portfolios. The estimated betas are less accurate for these small capitalization portfolios, and they suffer from a downward bias. Specifically, for the smallest decile firms the average leverage during the estimation period is only about 65% (= 1.253/1.846) of the leverage at the end of the estimation period, and this effect is (at least partially) responsible for the size effect. Prior market returns are not solely responsible for the correlation between size and leverage. Even within a given size decile, leverage impacts beta estimation and pricing errors. As a robustness test, we split the smallest size decile portfolio into ten subdecile portfolios according to their market value of equity. As shown in Table 3, even within this subsample of bad performing stocks, the size effect is observable and equity returns decrease with increasing size. Again, the size effect is most pronounced for the smallest subdecile portfolio. This portfolio has a high excess return of roughly 1.9% per month, which is 66.3 basis points higher than expected based on their estimated β 60 values. These findings are in line with the results in Fama and French (2007) that the size effect is caused by the migration of extremely small firms (microcaps) with extremely high returns. Knez and Ready (1997) also document that the size effect is caused by a few extreme observations. [Insert Table 3 and Figure 5 here] Analyzing the smallest subdecile portfolios in Table 3, we observe that the spread between the average leverage at the end of the estimation window, ØL t, and the average leverage within the estimation window, ØL 60, is even larger than in Table 2, which causes a more pronounced size effect. The estimated β 60 values are not able to explain the cross-section of excess returns across subdeciles. As illustrated in Figure 5, there is almost no relationship between β 60 and excess returns. While excess returns increase from 1.287% to 1.903% per month, the estimated betas increase only slightly from to Computing point-in-time betas The spread between L 60 and L t could be reduced by choosing a shorter beta estimation window. There is a trade-off, however, as shorter estimation windows will also lead to greater estimation errors in market model regressions. We propose a more convenient 17

19 solution to reduce pricing errors. By computing the point-in-time beta from equation (4) for each simulated firm, we correct the estimated betas for the bias due to the spread between ØL 60 and ØL t, thereby enhancing estimation accuracy by incorporating firms leverage at the end of the period. Table 2 reports the simulation results using our proposed correction of the endogenous leverage changes during the beta estimation window, and Figure 4 illustrates these findings. While for large firms the corrected betas are similar to the standard betas, the deviations are substantial for small cap stocks. The point-in-time betas, β t, lie well above the estimated betas, β 60, for the two smallest decile portfolios (1.693 versus and versus 1.262, respectively), implying that the standard beta estimation method causes a downward bias for the expected return on these portfolios. This downward bias is able to explain the size anomaly. In fact, using the point-in-time betas corrects for the bias, and the abnormal returns for the small size decile portfolios disappear. The pricing error, denoted as P E(β t ), becomes small and statistically insignificant even for the two smallest size decile portfolios (0.001 versus and versus 0.058, respectively). The effect of our beta correction is even more pronounced for subdeciles of the smallest portfolio, as shown in Figure 5. For these microcap stocks the standard beta has little explanatory power, while the point-in-time beta is able to match the observed returns. An exception is the smallest subdecile, where the expected return based on the pointin-time beta is even higher than the observed return. An explanation could be that equations (1) and (2) are correct only in continuous time, and using them to forecast discrete period returns leads to biased results because it implicitly assumes that the equity beta in equation (2) remains constant during the return measurement period (following the beta estimation window). Our simulations indicate that this inaccuracy does not lead to a notable bias as long as the time period is short enough and the level of leverage is not too high. For highly leveraged firms, however, the point-in-time beta will not remain constant during the period for which expected returns are estimated. If a firm does not default, its equity value sharply increases, the level of leverage decreases, and equation (4) will deliver an overestimated point-in-time beta for the smallest firms. 4.3 Discussion of simulation results Many explanations for the size effect have been discussed in the prior literature. Our approach is novel in that it does not introduce any frictions to the CAPM, such as trading costs, liquidity constraints, or irrational investors. We also omit adding new risk factors, thus the covariance with the market portfolio is the only source of return. 18

20 Furthermore, we do not require any new fitting parameters to align ex-ante and ex-post returns. Our findings indicate that even in a frictionless mean-variance setup, in which the CAPM holds unconditionally for asset returns, a size effect is observable if the beta is measured over longer time windows and does not incorporate the point-in-time leverage of the firm. Based on our simulation results, we suggest that the estimated pre-ranking betas must be corrected for changes in leverage during the beta estimation window. This correction is particularly important for the betas of small stocks. Nevertheless, this effect needs not be, or is even unlikely to be, the only size-related factor that affects stock returns. There are market frictions such as liquidity constraints, trading costs, information costs, estimation risk, and other factors which may affect the return the marginal investor requires to hold small capitalization firms in diversified portfolios. Our simulation setup indicates that none of these frictions are a necessary condition for a size effect to be observable when the standard beta estimation method is applied. Our results also have implications for empirical corporate finance research. Any test of a firm s risk-adjusted long-run stock market performance after some financing event has taken place simultaneously tests the validity of the expected return model ( bad-model problem ; Fama (1998)). Given that the beta from regressing a firm s stock returns on market returns over a given estimation period is a biased estimate for the point-in-time beta, using the standard beta will produce biased results. For example, if an event study identifies two subsamples of positive and negative post-event returns, the average beta of the positive return subsample will decrease, whereas the average beta of the negative return subsample will increase over time. In the standard event study methodology, the benchmark fails to incorporate these changes, which in turn will lead to biased abnormal return estimates. Even using the point-in-time beta would not be accurate in long-run performance studies because the expected point-in-time beta will change itself after the event due to return-induced changes in leverage. Finally, whenever a firm actively changes its capital structure, its point-in-time beta will change as well. Analyzing capital structure changes in an event-study framework, e.g., the effect of share repurchases on subsequent stock returns, without adjusting the estimated betas appropriately is likely to lead to biased results. Investigating the effect of share repurchases without an adjustment for the resulting leverage increase, one would expect that the estimated betas are biased downwards. The resulting bias will be higher for firms that experience prior negative performance. For example, Ikenberry et al. (1995) document high abnormal returns for value stocks and neutral performance for 19

21 growth stocks after the repurchase. They attribute this effect to undervaluation prior to the event. While their argument remains perfectly valid, we suggest that part of the effect they observe may be attributable to a leverage-induced mismeasurement in stock market betas. 4.4 Simulation results for post-ranking betas If post-ranking betas are used to calculate expected returns, a similar size-effect is observable. As shown in Table 4, the post-ranking betas range from for the largest size decile portfolio to for the smallest decile portfolio. For the smallest size decile portfolio, this implies a pricing error of 0.272% per month. The bias is smaller compared to pre-ranking betas, but the t-value still indicates significance at the 1% level. Using post-ranking betas to calculate expected returns reduces the pricing error, but does not remove it. Therefore, studies that analyze return estimates based on post-ranking betas will find seemingly unexplained high returns for small stocks, although the effect is smaller compared to the use of pre-ranking betas. [Insert Table 4 and Figure 6 here] For the smallest size decile portfolio, we find a downward trend in equity values after portfolio formation. The average equity at ranking is 1.76 (E t ), but it falls to 1.42 (E 60 ) during the following beta estimation window. Despite this decrease in the average equity value, there is a strong downward trend in the average leverage after portfolio formation. For the smallest size decile portfolio, the average leverage is (ØL t ) at formation time, but it drops to an average of (ØL 60 ) during the following beta estimation window. Because the trend in equity values cannot explain the systematic change in leverage, we conclude that it is caused by the correlation between default and leverage. The average default rate of 0.48% per month indicates a material reduction of portfolio size during the beta estimation window, and defaulting firms will have a systematic bias towards higher leverage compared to the corresponding decile portfolio. Removing defaulted firms will, on average, result in a lower leverage of the remaining portfolio firms. The leverage mismatch between return measurement and (post-formation) beta estimation leads to biased betas and a size effect that is similar to the use of pre-ranking betas. Finally, Figure 6 shows that the leverage correction in equation (4) applied on post-ranking betas delivers bias-free point-in-time betas that closely track our simulated returns. The pricing error becomes insignificant for all decile portfolios. 20