I examine the two components of default risk and how they relate to stock returns, size, and book-to-market. High default risk firms do not necessarily have high levels of systematic asset risk. I show that the two components of default risk, asset volatility and leverage, are negatively related. I provide evidence that leverage differences across firms are not reflected in equity betas. Therefore, I construct firm returns using estimates of firm’s debt returns. The results indicate that a large part of the value premium and some of the size premium can be explained by differences in leverage across firms.
In this paper, I examine the relationship between the components of default risk and how they relate to the size and value factors in stock returns. The size and value factors are robust empirical factors for pricing stock returns. These factors, in addition to the market factor, lead to a three-factor asset pricing model. The most popular explanation for a multi-factor asset pricing model is time-varying risk and risk premia. In a multi- period model, an unconditional expression of risk will lead to multiple factors. Many studies including Fama and French, argue that the size and value factors are related to relative economic distress not captured by beta [
Typically, economic distress is associated with the risk of default. Vassalou and Xing argue that most of the size premium and some of the value premium are closely related to the default risk of a firm [
Several recent studies provide evidence that there is in fact a positive premium on the systematic component of default risk in stock returns (e.g. Kapadia; Chava and Purnanandam; Anginer and Yildizhan; Friewald, Wagner and Zechner) [
I argue that the components of default risk are directly related to the size and value premium. I argue that the separation of asset volatility and leverage is an important step when sorting out the relationship between size, book-to-market, and default risk. Leverage is mechanically related to the risk of equity. Covariance with the market or beta should mechanically increase as leverage increases (Hamada) [
Since beta is not capturing leverage, one could directly control for leverage in equity return asset pricing tests. However, I show that leverage choice is endogenously related to asset volatility, which is also shown by George and Hwang [
The second component of default risk, asset volatility, is not as clearly related to systematic risk of equity as leverage. I examine how asset volatility is linked to the known size and book-to-market ratios and role in returns after controlling for capital structure differences across firms. Firm size is negatively related to asset volatility and to default risk. Other studies have argued that the default risk premium in equity returns is driven by size (George and Hwang; Da and Gao) [
Since beta is measured with error, it is important to understand if the error is due to dynamic risk or some other form of measurement error not related to dynamic risk. I do not argue that the systematic risk of a firm is constant. In fact, if leverage changes over time, risk changes over time. In addition, the underlying dynamics of firm value (asset risk) may co-vary with the market in a different manner over time. I do argue that it is important to properly control for simple differences in leverage before deriving a complicated story about risk and risk premia dynamics. A contemporaneous paper by Choi examines firm returns and ties this to analyses of capital structure dynamics and firm risk dynamics [
I calculate firm returns in a different manner than Choi which allows for a larger cross-section and time-series of firms [
The remainder of the paper is organized as follows. First, I discuss the data used in the analysis. Second, I examine the relationship between beta, leverage, asset volatility, default risk, size, book-to-market, and stock returns. Third, I discuss the estimation of debt returns. Finally, I examine the differences in the cross-section of equity and firm returns and how they relate to default risk, size, and book-to-market.
I use several sources of data in this study. All accounting data are from the COMPUSTAT annual file. I use the CRSP monthly stock file for equity returns. When there is a delisting event, I use the last available monthly return from the CRSP delisting file to calculate returns. I merge the CRSP and COMPUSTAT data using the link file from the merged database, which is based on CRSP permanent number and COMPUSAT GVKEY. I exclude financial firms and insurance companies from the analysis, following the prior literature. I use monthly bond returns from Reuters EJV. These data are from 2001-2005 and cover almost all traded US corporate debt.
In addition to the return and accounting data, I also use data from Moody’s KMV (MKMV). The MKMV data is linked to CRSP and COMPUSTAT using GVKEY which is provided with the data. The data from MKMV includes an estimate of an annualized Expected Default Frequency (EDF). This is the measure of default risk I use throughout the paper. Moody’s KMV uses a proprietary model to estimate a distance-to-default. This structural model of default risk includes claims to multiple types of debt instruments and preferred stock. The probability of default or EDF credit measure is based on the historical distribution of MKMV’s measure of distance-to-default and default rates. In the process of estimating a distance-to-default, MKMV estimates a measure of volatility of the firms underlying asset returns. This is done by using an iterative procedure using equity return volatility information and the structural model formulas. The two formulas relate how asset volatility (returns) and equity volatility (returns) are related. I use this estimate of asset return volatility to control for asset risk when examining leverage and beta in the cross-section6.
One issue with the MKMV data is the release of new models throughout my sample period. The EDF 8 model was established in 2006. There were some minor adjustments to the underlying model, but the major adjustment was to the empirical mapping from distance-to-default to EDF. The new mapping included more default data and a new range for the EDF credit measures7. For the majority of the analysis, I use the EDF 7 model, which restricts the data to the end of 2005. There is no reliable data for EDF 7 or 8 prior to 1970, so the sample begins then. For the firm return analysis, I use the EDF 8 model to allow for the largest sample possible. This also ensures that the analysis is not sensitive to any particular calibration of the EDF model.
When estimating beta and estimating the debt returns, I use multiple return indices. The returns on all government bonds are from the CRSP monthly government bond file. The risk-free rate is the thirty day Treasury bill rate from Federal Reserve statistical release. Corporate bond return indices are from two sources: Ibbotson Associates (pre-1989) and Lehman Brothers/Barclays Capital (post-1989). I have a complete time-series for five subsets of long-term corporate bonds: Aaa, Aa, A, Baa, and high- yield (low-grade). The Ibbotson high-yield index includes bonds rated below Baa. There is no breakdown between Ba, B, and Caa ratings. Therefore, I average the returns for the Ba, B, and Caa indices from the Lehman Brothers/Barclays Capital data to create a consistent high-yield index throughout the sample.
The correlation structure of the indices is similar to that found in Cornell and Green and other previous studies [
Change in the isk-Free Rate | Aaa Corporate Bond Excess Return | Aa Corporate Bond Excess Return | A Corporate Bond Excess Return | Baa Corporate Bond Excess Return | Low Grade Corporate Bond Excess Return | Value -Weighted CRSP Stock Excess Return | Equal -Weighted CRSP Stock Excess Return | 30 Year CRSP Government Bond Excess Return | 10 Year CRSP Government Bond Excess Return | 1 Year CRSP Government Bond Excess Return | |
---|---|---|---|---|---|---|---|---|---|---|---|
1970:1-2011:12 | |||||||||||
Change in the Risk-Free Rate | 1 | −0.08 | −0.10 | −0.14 | −0.16 | −0.11 | −0.14 | −0.16 | −0.04 | −0.05 | −0.16 |
Aaa Corporate Bond Excess Return | −0.08 | 1 | 0.92 | 0.89 | 0.80 | 0.41 | 0.27 | 0.17 | 0.81 | 0.83 | 0.44 |
Aa Corporate Bond Excess Return | −0.10 | 0.92 | 1 | 0.94 | 0.85 | 0.48 | 0.32 | 0.25 | 0.73 | 0.77 | 0.43 |
A Corporate Bond Excess Return | −0.14 | 0.89 | 0.94 | 1 | 0.91 | 0.52 | 0.35 | 0.29 | 0.73 | 0.75 | 0.38 |
Baa Corporate Bond Excess Return | −0.16 | 0.80 | 0.85 | 0.91 | 1 | 0.60 | 0.44 | 0.40 | 0.61 | 0.66 | 0.29 |
Low Grade Corporate Bond Excess Return | −0.11 | 0.41 | 0.48 | 0.52 | 0.60 | 1 | 0.54 | 0.59 | 0.17 | 0.24 | 0.09 |
Value -Weighted CRSP Stock Excess Return | −0.14 | 0.27 | 0.32 | 0.35 | 0.44 | 0.54 | 1 | 0.86 | 0.10 | 0.10 | −0.09 |
Equal -Weighted CRSP Stock Excess Return | −0.16 | 0.17 | 0.25 | 0.29 | 0.40 | 0.59 | 0.86 | 1 | 0.00 | −0.01 | −0.05 |
30 Year CRSP Government Bond Excess Return | −0.04 | 0.81 | 0.73 | 0.73 | 0.61 | 0.17 | 0.10 | 0.00 | 1 | 0.89 | 0.46 |
10 Year CRSP Government Bond Excess Return | −0.05 | 0.83 | 0.77 | 0.75 | 0.66 | 0.24 | 0.10 | −0.01 | 0.89 | 1 | 0.55 |
1 Year CRSP Government Bond Excess Return | −0.16 | 0.44 | 0.43 | 0.38 | 0.29 | 0.09 | −0.09 | −0.05 | 0.46 | 0.55 | 1 |
1970:1-1989:12 | |||||||||||
Change in the Risk-Free Rate | 1 | −0.11 | −0.15 | −0.21 | −0.25 | −0.22 | −0.21 | −0.21 | −0.06 | −0.08 | −0.24 |
Aaa Corporate Bond Excess Return | −0.11 | 1 | 0.91 | 0.89 | 0.85 | 0.67 | 0.37 | 0.25 | 0.85 | 0.85 | 0.60 |
Aa Corporate Bond Excess Return | −0.15 | 0.91 | 1 | 0.93 | 0.83 | 0.69 | 0.40 | 0.31 | 0.78 | 0.78 | 0.60 |
A Corporate Bond Excess Return | −0.21 | 0.89 | 0.93 | 1 | 0.87 | 0.70 | 0.43 | 0.35 | 0.76 | 0.76 | 0.55 |
Baa Corporate Bond Excess Return | −0.25 | 0.85 | 0.83 | 0.87 | 1 | 0.70 | 0.47 | 0.40 | 0.72 | 0.72 | 0.50 |
Low Grade Corporate Bond Excess Return | −0.22 | 0.67 | 0.69 | 0.70 | 0.70 | 1 | 0.54 | 0.50 | 0.53 | 0.56 | 0.40 |
Value -Weighted CRSP Stock Excess Return | −0.21 | 0.37 | 0.40 | 0.43 | 0.47 | 0.54 | 1 | 0.87 | 0.33 | 0.29 | 0.14 |
Equal -Weighted CRSP Stock Excess Return | −0.21 | 0.25 | 0.31 | 0.35 | 0.40 | 0.50 | 0.87 | 1 | 0.19 | 0.14 | 0.11 |
30 Year CRSP Government Bond Excess Return | −0.06 | 0.85 | 0.78 | 0.76 | 0.72 | 0.53 | 0.33 | 0.19 | 1 | 0.88 | 0.55 |
10 Year CRSP Government Bond Excess Return | −0.08 | 0.85 | 0.78 | 0.76 | 0.72 | 0.56 | 0.29 | 0.14 | 0.88 | 1 | 0.63 |
1 Year CRSP Government Bond Excess Return | −0.24 | 0.60 | 0.60 | 0.55 | 0.50 | 0.40 | 0.14 | 0.11 | 0.55 | 0.63 | 1 |
1989:1-2011:12 | |||||||||||
Change in the Risk-Free Rate | 1 | 0.03 | 0.03 | 0.03 | 0.05 | 0.03 | 0.01 | −0.05 | 0.02 | 0.03 | −0.11 |
Aaa Corporate Bond Excess Return | 0.03 | 1 | 0.94 | 0.88 | 0.76 | 0.19 | 0.14 | 0.05 | 0.79 | 0.81 | 0.32 |
Aa Corporate Bond Excess Return | 0.03 | 0.94 | 1 | 0.95 | 0.88 | 0.33 | 0.24 | 0.18 | 0.69 | 0.74 | 0.30 |
A Corporate Bond Excess Return | 0.03 | 0.88 | 0.95 | 1 | 0.94 | 0.40 | 0.27 | 0.23 | 0.70 | 0.75 | 0.26 |
Baa Corporate Bond Excess Return | 0.05 | 0.76 | 0.88 | 0.94 | 1 | 0.54 | 0.40 | 0.39 | 0.51 | 0.58 | 0.13 |
Low Grade Corporate Bond Excess Return | 0.03 | 0.19 | 0.33 | 0.40 | 0.54 | 1 | 0.56 | 0.68 | −0.08 | −0.03 | −0.08 |
Value -Weighted CRSP Stock Excess Return | 0.01 | 0.14 | 0.24 | 0.27 | 0.40 | 0.56 | 1 | 0.85 | −0.12 | −0.12 | −0.26 |
Equal -Weighted CRSP Stock Excess Return | −0.05 | 0.05 | 0.18 | 0.23 | 0.39 | 0.68 | 0.85 | 1 | −0.20 | −0.20 | −0.19 |
30 Year CRSP Government Bond Excess Return | 0.02 | 0.79 | 0.69 | 0.70 | 0.51 | −0.08 | −0.12 | −0.20 | 1 | 0.91 | 0.40 |
10 Year CRSP Government Bond Excess Return | 0.03 | 0.81 | 0.74 | 0.75 | 0.58 | −0.03 | −0.12 | −0.20 | 0.91 | 1 | 0.51 |
1 Year CRSP Government Bond Excess Return | −0.11 | 0.32 | 0.30 | 0.26 | 0.13 | −0.08 | −0.26 | −0.19 | 0.40 | 0.51 | 1 |
return on the CRSP value-weighted and equal-weighted indices, respectively. This correlation is higher in the later sub-sample. Since the high-grade bond indices are highly correlated, I do not include all of the corporate bond indices in the multi-beta specification. I equal weight the returns for the four high-grade indices and include a high-grade (investment-grade) and low-grade (speculative-grade) index. Innovations in the risk- free rate are negatively related to all the return indices for the full sample. Only the short-term government bond index and the equal-weighted CRSP stock index are negatively related in the recent sub-sample.
Since beta should capture any asset risk and leverage effect, I examine two questions regarding the relationship between equity beta and leverage. The first question concerns estimation error in equity beta from estimating the return on the market portfolio. The second question concerns the general cross-sectional relationship between leverage and equity beta after controlling for differences in asset volatility across firms.
To address the measurement error issue related to the market portfolio, I estimate equity beta using definitions of the market portfolio which include and exclude debt returns. Based on Ferguson and Shockley, measurement error in beta related to excluding debt claims from the market portfolio is correlated with financial leverage in the cross-section [
I estimate equity beta using the CRSP value-weighted stock market return and compare the results when bond indices are included. I estimate a single beta and a multi-beta regression, both including debt returns. For the multi-beta specification I assume the excess return on the five proxy portfolios spans the excess return on the true market portfolio. Several papers, including Shanken, use bond and stock portfolios as proxies for the market portfolio [
I use two different weighting schemes to compute the market return for (3). Ferguson and Shockley argue that the inclusion of low-grade bond returns is especially important in beta estimation due to the higher covariance with equity returns [
I use five-year rolling regressions to measure beta. For each five-year period, I require at least two years of data for a beta estimate. For comparability with Fama and French, I also estimate a post-ranking beta using pre-ranking beta and size sorted portfolios [
I measure leverage using different definitions in the analysis. Theory says the market value of leverage is the important measure when it comes to grossing up systematic risk. However, the market value of the firm is not directly observable. To overcome this issue, I use the result of the structural model equations to estimate the value of the firm. MKMV uses the estimated market value of the firm to compute asset volatility and then iteratively solve for the solution of the model. Therefore, I calculate market leverage as the market value of the firm minus the market value of equity divided by the market value of the firm.
I also compute a measure of leverage using the book value of debt as a proxy for the market value. This typically will understate the value of debt and leverage (Sweeny, Warga, and Winters) [
I skip six months between the first stock return and the end of the accounting fiscal year to be consistent with Fama and French [
The average level of asset volatility may describe part of the beta puzzle, but not all of it. Asset volatility is monotonically decreasing across the leverage deciles. The firms with the highest leverage have an average asset volatility of around 20%, while the firms with the lowest leverage have an asset volatility of around 45%. This is a large difference in volatility, but this may not translate to the asset beta. EDF is increasing across the leverage deciles as expected, but the relationship is muted by the offsetting asset volatility relationship. The two components of default risk are moving in opposite directions leading to offsetting effects, but the leverage effect seems to dominate the overall impact on EDF. That is, firms with high leverage and low asset volatility have higher average default risk than firms with low leverage and high asset volatility.
Panel D contains average values of the key variables sorted by EDF. The two components of the EDF are both increasing across the EDF deciles. Asset volatility only increases marginally from 24% to 33%, but market leverage increases from around 18% to 60%. All beta measures are increasing in EDF which is inconsistent with the Campbell et al. results [
The data in Panel E and F are based on decile ranks using two beta measures to judge the impact of overweighting the debt claims in the market portfolio. Panel E data are based on the normal beta using the CRSP value-weighted index as the market portfolio. Panel F data are based on the VW1 market portfolio beta, which overweights low-grade debt. Leverage is negatively related to both beta measures, but the overweighting reduces
Panel A | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Market Leverage | Market Leverage | Book Debt Leverage | Book Assets Leverage | Beta-CRSP VW Only | Beta-VW1 | Beta-VW2 | Multi-beta LG Beta | Post-ranking beta | Empirical Asset Vol. | EDF (%) | Stock Price | LN(ME) | LN(BE/ME) |
Low Leverage | 0.13 | 0.03 | 0.05 | 1.23 | 1.90 | 1.80 | 0.30 | 1.31 | 0.42 | 0.73 | 22.96 | 4.77 | −1.02 |
2 | 0.16 | 0.05 | 0.10 | 1.35 | 2.08 | 1.97 | 0.28 | 1.33 | 0.42 | 0.71 | 25.40 | 5.26 | −1.16 |
3 | 0.21 | 0.10 | 0.16 | 1.28 | 2.01 | 1.88 | 0.30 | 1.30 | 0.37 | 0.82 | 25.27 | 5.28 | −0.98 |
4 | 0.26 | 0.15 | 0.22 | 1.20 | 1.88 | 1.76 | 0.28 | 1.26 | 0.33 | 0.89 | 25.23 | 5.35 | −0.83 |
5 | 0.33 | 0.22 | 0.28 | 1.15 | 1.81 | 1.68 | 0.25 | 1.24 | 0.29 | 1.04 | 24.81 | 5.33 | −0.68 |
6 | 0.39 | 0.29 | 0.33 | 1.11 | 1.75 | 1.62 | 0.25 | 1.22 | 0.27 | 1.21 | 23.40 | 5.24 | −0.53 |
7 | 0.45 | 0.37 | 0.38 | 1.08 | 1.72 | 1.58 | 0.27 | 1.21 | 0.24 | 1.40 | 21.75 | 5.08 | −0.39 |
8 | 0.52 | 0.46 | 0.43 | 1.03 | 1.67 | 1.52 | 0.26 | 1.19 | 0.21 | 1.64 | 20.23 | 4.98 | −0.25 |
9 | 0.60 | 0.57 | 0.50 | 0.93 | 1.59 | 1.41 | 0.26 | 1.13 | 0.18 | 1.88 | 18.80 | 4.97 | −0.12 |
High Leverage | 0.70 | 0.72 | 0.57 | 1.03 | 1.75 | 1.55 | 0.36 | 1.23 | 0.18 | 3.76 | 15.30 | 4.38 | 0.29 |
High-Low | 0.57 | 0.69 | 0.52 | −0.20 | −0.15 | −0.25 | 0.06 | −0.08 | −0.24 | 3.03 | −7.66 | −0.39 | 1.31 |
Panel B | |||||||||||||
Book Debt Leverage | |||||||||||||
Low Leverage | 0.11 | 0.01 | 0.03 | 1.37 | 2.10 | 2.00 | 0.33 | 1.37 | 0.46 | 0.68 | 23.51 | 4.94 | −1.30 |
2 | 0.16 | 0.05 | 0.10 | 1.31 | 2.03 | 1.91 | 0.27 | 1.32 | 0.41 | 0.73 | 24.31 | 5.14 | −1.09 |
3 | 0.21 | 0.10 | 0.17 | 1.23 | 1.93 | 1.80 | 0.29 | 1.28 | 0.36 | 0.85 | 24.45 | 5.17 | −0.91 |
4 | 0.27 | 0.15 | 0.22 | 1.18 | 1.86 | 1.73 | 0.28 | 1.26 | 0.32 | 0.96 | 24.67 | 5.23 | −0.77 |
5 | 0.33 | 0.22 | 0.28 | 1.14 | 1.79 | 1.66 | 0.26 | 1.24 | 0.29 | 1.10 | 23.69 | 5.21 | −0.64 |
6 | 0.39 | 0.29 | 0.32 | 1.11 | 1.75 | 1.62 | 0.26 | 1.23 | 0.26 | 1.26 | 23.14 | 5.16 | −0.50 |
7 | 0.46 | 0.37 | 0.37 | 1.08 | 1.72 | 1.59 | 0.28 | 1.22 | 0.24 | 1.48 | 21.88 | 5.04 | −0.35 |
8 | 0.53 | 0.47 | 0.43 | 1.04 | 1.67 | 1.53 | 0.26 | 1.20 | 0.21 | 1.73 | 20.71 | 5.00 | −0.23 |
9 | 0.60 | 0.57 | 0.51 | 0.92 | 1.57 | 1.39 | 0.25 | 1.12 | 0.18 | 1.89 | 19.71 | 5.02 | −0.11 |
High Leverage | 0.70 | 0.73 | 0.59 | 0.98 | 1.71 | 1.49 | 0.34 | 1.17 | 0.17 | 3.44 | 16.74 | 4.63 | 0.29 |
High-Low | 0.59 | 0.72 | 0.56 | −0.39 | −0.39 | −0.51 | 0.01 | −0.19 | −0.29 | 2.76 | −6.77 | −0.32 | 1.59 |
Panel C | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Book Assets Leverage | |||||||||||||
Low Leverage | 0.06 | 0.04 | 0.09 | 1.33 | 2.04 | 1.95 | 0.32 | 1.32 | 0.47 | 0.28 | 26.84 | 5.29 | −1.48 |
2 | 0.13 | 0.08 | 0.14 | 1.29 | 1.99 | 1.88 | 0.26 | 1.30 | 0.40 | 0.50 | 25.93 | 5.28 | −1.06 |
3 | 0.20 | 0.12 | 0.19 | 1.24 | 1.94 | 1.82 | 0.27 | 1.29 | 0.36 | 0.70 | 24.69 | 5.20 | −0.85 |
4 | 0.26 | 0.17 | 0.23 | 1.19 | 1.88 | 1.74 | 0.27 | 1.26 | 0.33 | 0.88 | 23.86 | 5.18 | −0.71 |
5 | 0.33 | 0.23 | 0.28 | 1.15 | 1.82 | 1.68 | 0.27 | 1.24 | 0.30 | 1.07 | 23.34 | 5.15 | −0.58 |
6 | 0.40 | 0.30 | 0.32 | 1.11 | 1.76 | 1.63 | 0.25 | 1.23 | 0.27 | 1.28 | 22.20 | 5.11 | −0.45 |
7 | 0.47 | 0.37 | 0.36 | 1.07 | 1.72 | 1.57 | 0.26 | 1.21 | 0.24 | 1.51 | 21.13 | 5.04 | −0.34 |
8 | 0.54 | 0.45 | 0.42 | 1.00 | 1.64 | 1.48 | 0.26 | 1.17 | 0.21 | 1.70 | 20.09 | 5.00 | −0.23 |
9 | 0.62 | 0.54 | 0.47 | 0.94 | 1.59 | 1.42 | 0.27 | 1.14 | 0.18 | 2.05 | 18.80 | 4.92 | −0.09 |
High Leverage | 0.75 | 0.67 | 0.52 | 1.04 | 1.75 | 1.55 | 0.36 | 1.25 | 0.16 | 4.15 | 15.96 | 4.38 | 0.19 |
High-Low | 0.69 | 0.63 | 0.43 | −0.29 | −0.30 | −0.39 | 0.04 | −0.07 | −0.32 | 3.87 | −10.88 | −0.91 | 1.67 |
Panel D | |||||||||||||
MKMV Default Probability (EDF) | |||||||||||||
Low EDF | 0.18 | 0.14 | 0.22 | 0.87 | 1.37 | 1.28 | 0.05 | 0.98 | 0.24 | 0.06 | 44.32 | 6.84 | −1.17 |
2 | 0.27 | 0.23 | 0.29 | 0.95 | 1.52 | 1.40 | 0.12 | 1.05 | 0.25 | 0.15 | 32.53 | 6.32 | −0.87 |
3 | 0.31 | 0.25 | 0.30 | 1.01 | 1.62 | 1.49 | 0.19 | 1.11 | 0.26 | 0.25 | 28.09 | 5.90 | −0.75 |
4 | 0.33 | 0.26 | 0.30 | 1.06 | 1.70 | 1.57 | 0.23 | 1.16 | 0.27 | 0.39 | 24.55 | 5.51 | −0.67 |
5 | 0.35 | 0.28 | 0.29 | 1.12 | 1.78 | 1.64 | 0.25 | 1.22 | 0.29 | 0.56 | 21.61 | 5.17 | −0.60 |
6 | 0.38 | 0.29 | 0.30 | 1.17 | 1.87 | 1.72 | 0.31 | 1.28 | 0.30 | 0.80 | 18.90 | 4.83 | −0.52 |
7 | 0.40 | 0.31 | 0.30 | 1.22 | 1.95 | 1.80 | 0.36 | 1.33 | 0.31 | 1.13 | 16.68 | 4.53 | −0.45 |
8 | 0.44 | 0.34 | 0.31 | 1.27 | 2.03 | 1.87 | 0.41 | 1.37 | 0.32 | 1.64 | 14.45 | 4.23 | −0.35 |
9 | 0.49 | 0.38 | 0.33 | 1.32 | 2.11 | 1.95 | 0.41 | 1.43 | 0.33 | 2.58 | 11.92 | 3.87 | −0.25 |
High EDF | 0.60 | 0.48 | 0.37 | 1.36 | 2.18 | 2.00 | 0.45 | 1.48 | 0.33 | 6.57 | 9.81 | 3.36 | 0.03 |
High-Low | 0.42 | 0.34 | 0.15 | 0.48 | 0.81 | 0.72 | 0.39 | 0.50 | 0.09 | 6.51 | −34.50 | −3.48 | 1.20 |
Panel E | |||||||||||||
BETA-CRSP VW Only | |||||||||||||
Low Beta | 0.44 | 0.38 | 0.37 | 0.22 | 0.48 | 0.35 | 0.18 | 0.76 | 0.22 | 1.18 | 21.38 | 4.64 | −0.39 |
2 | 0.41 | 0.34 | 0.34 | 0.53 | 0.92 | 0.80 | 0.19 | 0.87 | 0.21 | 0.99 | 24.83 | 5.04 | −0.41 |
3 | 0.39 | 0.31 | 0.31 | 0.71 | 1.17 | 1.05 | 0.23 | 0.99 | 0.23 | 1.03 | 25.64 | 5.21 | −0.46 |
4 | 0.38 | 0.30 | 0.30 | 0.86 | 1.38 | 1.26 | 0.24 | 1.08 | 0.24 | 1.07 | 25.43 | 5.23 | −0.49 |
5 | 0.38 | 0.30 | 0.30 | 1.00 | 1.58 | 1.46 | 0.26 | 1.17 | 0.26 | 1.15 | 24.54 | 5.25 | −0.51 |
6 | 0.38 | 0.30 | 0.30 | 1.14 | 1.80 | 1.67 | 0.27 | 1.25 | 0.27 | 1.28 | 23.66 | 5.25 | −0.54 |
7 | 0.38 | 0.29 | 0.29 | 1.29 | 2.05 | 1.90 | 0.29 | 1.34 | 0.30 | 1.46 | 22.15 | 5.18 | −0.58 |
8 | 0.36 | 0.27 | 0.28 | 1.49 | 2.35 | 2.18 | 0.33 | 1.46 | 0.33 | 1.68 | 20.05 | 5.05 | −0.64 |
9 | 0.34 | 0.25 | 0.26 | 1.76 | 2.75 | 2.59 | 0.37 | 1.63 | 0.38 | 1.91 | 18.47 | 4.93 | −0.70 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
High Beta | 0.31 | 0.22 | 0.25 | 2.37 | 3.66 | 3.48 | 0.43 | 1.86 | 0.46 | 2.36 | 16.65 | 4.75 | −0.88 |
High-Low | −0.13 | −0.16 | −0.12 | 2.15 | 3.18 | 3.13 | 0.26 | 1.10 | 0.25 | 1.18 | −4.73 | 0.10 | −0.49 |
Panel F | |||||||||||||
BETA VW1-Large Low Grade Bond Weight in Market Portfolio | |||||||||||||
Low Beta | 0.39 | 0.30 | 0.30 | 0.40 | 0.18 | 0.38 | 0.02 | 0.89 | 0.26 | 1.46 | 20.64 | 4.37 | −0.50 |
2 | 0.40 | 0.32 | 0.32 | 0.63 | 0.81 | 0.84 | 0.09 | 0.94 | 0.23 | 1.04 | 24.57 | 5.03 | −0.47 |
3 | 0.40 | 0.33 | 0.33 | 0.75 | 1.10 | 1.06 | 0.12 | 1.00 | 0.23 | 0.98 | 25.51 | 5.27 | −0.46 |
4 | 0.40 | 0.32 | 0.33 | 0.87 | 1.34 | 1.26 | 0.17 | 1.07 | 0.24 | 1.01 | 25.50 | 5.38 | −0.48 |
5 | 0.39 | 0.31 | 0.31 | 1.00 | 1.57 | 1.45 | 0.22 | 1.15 | 0.25 | 1.13 | 24.91 | 5.33 | −0.50 |
6 | 0.38 | 0.30 | 0.30 | 1.13 | 1.81 | 1.66 | 0.25 | 1.23 | 0.27 | 1.23 | 24.06 | 5.30 | −0.53 |
7 | 0.37 | 0.29 | 0.29 | 1.27 | 2.07 | 1.89 | 0.31 | 1.32 | 0.29 | 1.39 | 22.72 | 5.24 | −0.56 |
8 | 0.37 | 0.28 | 0.29 | 1.45 | 2.40 | 2.18 | 0.38 | 1.43 | 0.32 | 1.63 | 20.48 | 5.08 | −0.60 |
9 | 0.36 | 0.27 | 0.28 | 1.69 | 2.87 | 2.57 | 0.47 | 1.58 | 0.36 | 1.91 | 18.48 | 4.91 | −0.65 |
High Beta | 0.32 | 0.24 | 0.26 | 2.17 | 3.99 | 3.43 | 0.76 | 1.78 | 0.44 | 2.34 | 15.96 | 4.63 | −0.83 |
High-Low | −0.06 | −0.06 | −0.04 | 1.77 | 3.81 | 3.05 | 0.74 | 0.88 | 0.18 | 0.88 | −4.68 | 0.26 | −0.33 |
this negative relationship slightly. Asset volatility is increasing as beta increases in both Panel E and F. The evidence here helps shed light on the puzzling beta and leverage relationship. The highest average beta is 2.37, with a corresponding volatility of 46%. The lowest average beta is 0.22, with a corresponding volatility of 22%. The beta does not seem to be purely related to the combination of asset volatility and leverage, which suggests asset volatility is not the best proxy for asset beta, or that there is severe mismeasurement of beta. Companies with the highest and lowest beta tend to be smaller, but book-to-market is negatively related to beta.
Book-to-market is strongly negatively related to all measures of leverage, but stronger for market measures of leverage. Size is also negatively related to market leverage, but positively related to book leverage. Interestingly, asset volatility is negatively related to both size and book-to-market. Small firms with low book-to-market have higher levels of asset volatility.
Before turning to the stock return analysis, I attempt to control for asset volatility
Beta CRSP Only | Beta VW1 | Beta VW2 | CRSP Beta | HGCB Beta | LGCB Beta | LTGB Beta | STGB Beta | Post-Ranking Beta | Market Leverage | Book Debt Leverage | Book Assets Leverage | EDF | Empirical Asset Vol. | LN (BE/ME) | LN(ME) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Beta CRSP Only | 1 | 0.84 | 0.97 | 0.89 | −0.10 | 0.07 | 0.00 | 0.03 | 0.88 | −0.16 | −0.17 | −0.15 | 0.16 | 0.48 | −0.16 | −0.02 |
Beta VW1 | 0.84 | 1 | 0.94 | 0.59 | 0.01 | 0.24 | 0.07 | 0.04 | 0.73 | −0.09 | −0.10 | −0.07 | 0.13 | 0.36 | −0.11 | −0.01 |
Beta VW2 | 0.97 | 0.94 | 1 | 0.79 | −0.07 | 0.12 | 0.06 | 0.03 | 0.84 | −0.14 | −0.15 | −0.12 | 0.14 | 0.44 | −0.15 | 0.00 |
CRSP Beta | 0.89 | 0.59 | 0.79 | 1 | −0.23 | −0.14 | 0.05 | 0.10 | 0.79 | −0.18 | −0.19 | −0.17 | 0.14 | 0.46 | −0.16 | 0.00 |
HGCB Beta | −0.10 | 0.01 | −0.07 | −0.23 | 1 | −0.42 | −0.72 | −0.12 | −0.09 | 0.07 | 0.06 | 0.04 | −0.02 | −0.12 | 0.06 | −0.01 |
LGCB Beta | 0.07 | 0.24 | 0.12 | −0.14 | −0.42 | 1 | 0.15 | −0.04 | 0.11 | 0.01 | 0.01 | −0.01 | 0.08 | 0.10 | 0.01 | −0.14 |
LTGB Beta | 0.00 | 0.07 | 0.06 | 0.05 | −0.72 | 0.15 | 1 | −0.15 | −0.04 | −0.04 | −0.01 | 0.03 | −0.04 | 0.02 | −0.06 | 0.10 |
STGB Beta | 0.03 | 0.04 | 0.03 | 0.10 | −0.12 | −0.04 | −0.15 | 1 | 0.01 | −0.01 | −0.02 | −0.03 | −0.01 | 0.00 | 0.02 | 0.02 |
Post-Ranking Beta | 0.88 | 0.73 | 0.84 | 0.79 | −0.09 | 0.11 | −0.04 | 0.01 | 1 | −0.11 | −0.16 | −0.20 | 0.29 | 0.51 | −0.05 | −0.35 |
Market Leverage | −0.16 | −0.09 | −0.14 | −0.18 | 0.07 | 0.01 | −0.04 | −0.01 | −0.11 | 1 | 0.87 | 0.67 | 0.44 | −0.62 | 0.55 | −0.13 |
Book Debt Leverage | −0.17 | −0.10 | −0.15 | −0.19 | 0.06 | 0.01 | −0.01 | −0.02 | −0.16 | 0.87 | 1 | 0.83 | 0.34 | −0.55 | 0.55 | −0.08 |
Book Assets Leverage | −0.15 | −0.07 | −0.12 | −0.17 | 0.04 | −0.01 | 0.03 | −0.03 | −0.20 | 0.67 | 0.83 | 1 | 0.15 | −0.47 | 0.11 | 0.14 |
EDF | 0.16 | 0.13 | 0.14 | 0.14 | −0.02 | 0.08 | −0.04 | −0.01 | 0.29 | 0.44 | 0.34 | 0.15 | 1 | 0.11 | 0.31 | −0.42 |
Empirical Asset Vol. | 0.48 | 0.36 | 0.44 | 0.46 | −0.12 | 0.10 | 0.02 | 0.00 | 0.51 | −0.62 | −0.55 | −0.47 | 0.11 | 1 | −0.36 | −0.23 |
LN (BE/ME) | −0.16 | −0.11 | −0.15 | −0.16 | 0.06 | 0.01 | −0.06 | 0.02 | −0.05 | 0.55 | 0.55 | 0.11 | 0.31 | −0.36 | 1 | −0.35 |
LN(ME) | −0.02 | −0.01 | 0.00 | 0.00 | −0.01 | −0.14 | 0.10 | 0.02 | −0.35 | −0.13 | −0.08 | 0.14 | −0.42 | −0.23 | −0.35 | 1 |
and then sort by leverage to help understand the relationship between leverage and beta.
The beta and leverage relationship is now positive after conditioning on asset volatile-
Empirical Asset Volatility | Leverage | Empirical Asset Vol. | BETA-CRSP VW Only | BETA-High Low Grade Bond Weight | ln(BE/ME) | ln(ME) | EDF(%) | |
---|---|---|---|---|---|---|---|---|
Low Asset Volatility | Low Leverage | 0.37 | 0.14 | 0.67 | 1.12 | −0.49 | 6.17 | 0.24 |
2 | 0.52 | 0.13 | 0.68 | 1.22 | −0.34 | 5.91 | 0.47 | |
3 | 0.60 | 0.12 | 0.67 | 1.24 | −0.21 | 5.75 | 0.67 | |
4 | 0.67 | 0.12 | 0.75 | 1.35 | −0.07 | 5.35 | 1.12 | |
High Leverage | 0.80 | 0.11 | 0.97 | 1.64 | 0.23 | 4.51 | 3.45 | |
High-Low | 0.42 | −0.03 | 0.30 | 0.53 | 0.72 | −1.66 | 3.21 | |
Low Leverage | 0.20 | 0.20 | 0.76 | 1.24 | −0.81 | 6.31 | 0.13 | |
2 | 0.35 | 0.19 | 0.87 | 1.41 | −0.56 | 6.01 | 0.35 | |
3 | 0.45 | 0.19 | 0.96 | 1.54 | −0.37 | 5.56 | 0.66 | |
4 | 0.55 | 0.19 | 1.02 | 1.65 | −0.17 | 4.95 | 1.27 | |
High Leverage | 0.70 | 0.19 | 1.15 | 1.88 | 0.13 | 4.16 | 3.71 | |
High-Low | 0.49 | −0.01 | 0.39 | 0.65 | 0.94 | −2.15 | 3.59 | |
Low Leverage | 0.13 | 0.26 | 0.88 | 1.38 | −1.05 | 6.15 | 0.12 | |
2 | 0.25 | 0.26 | 0.99 | 1.58 | −0.69 | 5.61 | 0.36 | |
3 | 0.35 | 0.26 | 1.08 | 1.70 | −0.52 | 5.21 | 0.72 | |
4 | 0.46 | 0.25 | 1.15 | 1.85 | −0.33 | 4.66 | 1.42 | |
High Leverage | 0.62 | 0.25 | 1.27 | 2.05 | −0.01 | 3.99 | 4.09 | |
High-Low | 0.48 | −0.01 | 0.38 | 0.68 | 1.04 | −2.16 | 3.96 | |
Low Leverage | 0.09 | 0.36 | 1.11 | 1.70 | −1.26 | 5.65 | 0.15 | |
2 | 0.18 | 0.35 | 1.23 | 1.92 | −0.91 | 5.23 | 0.43 | |
3 | 0.27 | 0.35 | 1.27 | 2.00 | −0.71 | 4.76 | 0.87 | |
4 | 0.37 | 0.35 | 1.35 | 2.12 | −0.50 | 4.39 | 1.63 | |
High Leverage | 0.53 | 0.34 | 1.43 | 2.27 | −0.16 | 3.86 | 4.46 | |
High-Low | 0.44 | −0.01 | 0.32 | 0.57 | 1.10 | −1.79 | 4.31 | |
High Asset Volatility | Low Leverage | 0.04 | 0.58 | 1.55 | 2.37 | −1.70 | 4.95 | 0.32 |
2 | 0.10 | 0.55 | 1.65 | 2.54 | −1.26 | 4.67 | 0.68 | |
3 | 0.16 | 0.53 | 1.65 | 2.53 | −1.04 | 4.50 | 1.12 | |
4 | 0.24 | 0.51 | 1.63 | 2.52 | −0.81 | 4.23 | 1.89 | |
High Leverage | 0.40 | 0.49 | 1.65 | 2.56 | −0.43 | 3.79 | 4.84 | |
High-Low | 0.36 | −0.09 | 0.10 | 0.20 | 1.27 | −1.16 | 4.52 |
ity as expected. However the increase in beta is not large enough. The 35% - 50% increase in leverage only translates into an increase in beta of around 0.30 for the low-volatility firms and much less for the high-volatility firms. If we follow the simple formula presented in most text books for un-levering beta, the resulting difference should be much larger. This implies that beta is not picking up mechanical leverage effects due to the way it is measured.
To formally test the Ferguson and Shockley argument that missing debt claims from the market portfolio are driving the book-to-market and size effects through their correlation with leverage, I estimate Fama and MacBeth style regressions [
The key test is to measure the impact on the coefficients on size, book-to-market, and EDF after including betas measured with risky debt claims in the market portfolio. I adjust the standard errors for auto-correlation by regressing the time-series of the
ln(BE/ME) | ln(ME) | ln(EDF) | CRSP VW beta (No Bonds) | Post-ranking beta | CRSP VW1 beta (With Bonds) | CRSP VW beta | HG Corp. Bond beta | LG Corp. Bond beta | LT Gov. Bond beta | ST Gov. Bond beta | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Average Coefficient | 0.301% | −0.044% | ||||||||||
t-statistic | 3.23 | −1.08 | ||||||||||
Average Coefficient | 0.312% | −0.070% | −0.062% | |||||||||
t-statistic | 3.05 | −2.04 | −1.03 | |||||||||
Average Coefficient | 0.304% | −0.074% | −0.066% | 0.035% | ||||||||
t-statistic | 3.63 | −2.08 | −1.55 | 0.26 | ||||||||
Average Coefficient | 0.310% | −0.074% | −0.071% | 0.030% | ||||||||
t-statistic | 3.71 | −2.20 | −1.68 | 0.13 | ||||||||
Average Coefficient | 0.314% | −0.075% | −0.062% | 0.013% | ||||||||
t-statistic | 3.42 | −2.12 | −1.30 | 0.24 | ||||||||
Average Coefficient | 0.299% | −0.077% | −0.066% | 0.069% | 0.045% | 0.032% | 0.010% | 0.002% | ||||
t-statistic | 3.70 | −2.26 | −1.63 | 0.55 | 1.45 | 0.52 | 0.21 | 0.97 |
coefficients on a constant and use a six-lag Newey-West adjustment. I do not make a correction for estimation error in the betas in the first pass regression. The levels of the t-statistics are already low and the correction will likely cause the standard errors to increase further (Shanken) [
The average coefficient on book-to-market is 0.301% and more than three standard errors from zero. The average coefficient on the size is −0.044% and only one standard error from zero. The lack of significant size effect is driven by the deletion of firms with stock prices under $5.00. Unreported results show that the point estimates are similar to Fama and French when including all firms in the cross-sectional regressions [
The key is to examine the coefficients on size, book-to-market, and EDF without beta in the regression versus when beta is in the regression. I include four specifications in
I have established that the market portfolio definition is not the cause of the measurement error issue in beta. This leaves three possible explanations for the negative cross-sectional relationship between beta and leverage. First, it is possible that firms with high asset beta have low levels of leverage either by choice or not. Second, dynamic leverage in the short run may cause unconditional beta estimation errors. Third, some other form of measurement error in beta is causing beta to not reflect market leverage.
I have provided evidence in
If market leverage is dynamic in the short run, a five-year rolling estimate of beta will not capture the true equity beta of the firm. This would require leverage to change quickly at the end of the measurement period and be much different than the five-year average. To examine this further, I present average stock returns in
The average stock returns in Panel A of
If we examine the pattern in Panel D, it is consistent with the leverage deciles in Panel A and B. High EDF firms have large negative recent stock returns causing spikes in market leverage. However the pattern reverses at the month of portfolio formation. There is a small positive return in the first month of the portfolio formation after deleting small firms; however, this disappears in the next two months. Since I control for delisting returns, this is not likely to be explained purely by deleting firms that actually default in the month after portfolio formation.
The main question is how to proceed to disentangle these effects and how they relate to default risk. Choi argues that leverage is increasing rapidly for high book-to-market firms when the price of risk is high, but low book-to-market firms do not experience large movements in leverage [
I argue that book-to-market is picking up some of the issues related to dynamic leverage better than EDF. This is because of the components of default risk; book-to- market is positively related to leverage and negatively related to asset volatility. However, size is negatively related to both components. I argue that examining the cross-section of firm returns relative to equity returns is the best way to isolate the leverage impact,
Panel A | ||||||||
---|---|---|---|---|---|---|---|---|
Market Leverage | Stock Return t − 5 | Stock Return t − 4 | Stock Return t − 3 | Stock Return t − 2 | Stock Return t-1 | Stock Return t | Stock Return t + 1 | Stock Return t + 2 |
Low Leverage | 2.53% | 2.56% | 2.54% | 2.67% | 2.83% | 1.22% | 1.24% | 1.22% |
2 | 2.01% | 1.98% | 2.04% | 2.07% | 2.18% | 1.17% | 1.24% | 1.15% |
3 | 1.63% | 1.68% | 1.67% | 1.71% | 1.78% | 1.27% | 1.31% | 1.31% |
4 | 1.49% | 1.48% | 1.50% | 1.54% | 1.59% | 1.30% | 1.32% | 1.31% |
5 | 1.24% | 1.28% | 1.25% | 1.34% | 1.45% | 1.40% | 1.38% | 1.38% |
6 | 1.11% | 1.13% | 1.20% | 1.21% | 1.29% | 1.31% | 1.37% | 1.34% |
7 | 0.97% | 1.05% | 1.03% | 1.06% | 1.05% | 1.41% | 1.36% | 1.34% |
8 | 0.78% | 0.79% | 0.81% | 0.89% | 0.90% | 1.34% | 1.33% | 1.37% |
9 | 0.48% | 0.47% | 0.54% | 0.55% | 0.59% | 1.40% | 1.38% | 1.32% |
High Leverage | −0.62% | −0.56% | −0.53% | −0.52% | −0.56% | 1.57% | 1.42% | 1.37% |
High-Low | −3.15% | −3.11% | −3.07% | −3.19% | −3.39% | 0.34% | 0.18% | 0.15% |
Panel B | ||||||||
Book Debt Leverage | ||||||||
Low Leverage | 1.95% | 1.99% | 1.93% | 2.00% | 2.09% | 1.25% | 1.25% | 1.22% |
2 | 1.84% | 1.87% | 1.87% | 1.92% | 2.00% | 1.21% | 1.26% | 1.23% |
3 | 1.66% | 1.63% | 1.65% | 1.67% | 1.79% | 1.25% | 1.28% | 1.24% |
4 | 1.59% | 1.58% | 1.56% | 1.63% | 1.66% | 1.27% | 1.32% | 1.32% |
5 | 1.39% | 1.40% | 1.44% | 1.48% | 1.53% | 1.34% | 1.33% | 1.31% |
6 | 1.17% | 1.18% | 1.15% | 1.22% | 1.37% | 1.33% | 1.37% | 1.36% |
7 | 0.98% | 0.99% | 1.07% | 1.10% | 1.09% | 1.34% | 1.34% | 1.37% |
8 | 0.79% | 0.86% | 0.88% | 0.95% | 0.98% | 1.35% | 1.33% | 1.33% |
9 | 0.59% | 0.61% | 0.69% | 0.68% | 0.71% | 1.37% | 1.36% | 1.34% |
High Leverage | −0.34% | −0.26% | −0.19% | −0.13% | −0.12% | 1.68% | 1.51% | 1.41% |
High-Low | −2.29% | −2.25% | −2.13% | −2.13% | −2.21% | 0.43% | 0.26% | 0.19% |
Panel C | ||||||||
Book Assets Leverage | ||||||||
Low Leverage | 1.21% | 1.26% | 1.24% | 1.28% | 1.31% | 1.33% | 1.33% | 1.29% |
2 | 1.10% | 1.17% | 1.19% | 1.24% | 1.32% | 1.37% | 1.35% | 1.34% |
---|---|---|---|---|---|---|---|---|
3 | 1.32% | 1.28% | 1.31% | 1.37% | 1.39% | 1.38% | 1.39% | 1.35% |
4 | 1.24% | 1.26% | 1.26% | 1.27% | 1.36% | 1.38% | 1.38% | 1.35% |
5 | 1.24% | 1.26% | 1.27% | 1.32% | 1.40% | 1.42% | 1.39% | 1.37% |
6 | 1.17% | 1.19% | 1.24% | 1.30% | 1.39% | 1.42% | 1.40% | 1.38% |
7 | 1.17% | 1.17% | 1.23% | 1.31% | 1.37% | 1.40% | 1.40% | 1.38% |
8 | 1.07% | 1.10% | 1.12% | 1.18% | 1.21% | 1.27% | 1.30% | 1.26% |
9 | 1.04% | 1.09% | 1.10% | 1.11% | 1.22% | 1.24% | 1.25% | 1.26% |
High Leverage | 1.06% | 1.06% | 1.10% | 1.12% | 1.12% | 1.17% | 1.18% | 1.15% |
High-Low | −0.14% | −0.20% | −0.15% | −0.16% | −0.18% | −0.15% | −0.15% | −0.14% |
Panel D | ||||||||
MKMV Default Probability (EDF) | ||||||||
Low EDF | 2.10% | 2.10% | 2.13% | 2.22% | 2.38% | 1.13% | 1.19% | 1.20% |
2 | 2.09% | 2.10% | 2.15% | 2.18% | 2.31% | 1.17% | 1.19% | 1.20% |
3 | 2.06% | 2.07% | 2.09% | 2.14% | 2.25% | 1.26% | 1.30% | 1.29% |
4 | 1.97% | 1.96% | 1.93% | 2.02% | 2.12% | 1.34% | 1.36% | 1.35% |
5 | 1.80% | 1.81% | 1.86% | 1.86% | 1.94% | 1.43% | 1.47% | 1.47% |
6 | 1.59% | 1.63% | 1.60% | 1.72% | 1.75% | 1.36% | 1.44% | 1.35% |
7 | 1.40% | 1.33% | 1.29% | 1.33% | 1.44% | 1.42% | 1.50% | 1.42% |
8 | 0.81% | 0.83% | 0.86% | 0.88% | 0.87% | 1.49% | 1.37% | 1.37% |
9 | 0.04% | 0.05% | 0.09% | 0.12% | 0.10% | 1.29% | 1.34% | 1.30% |
High EDF | −2.24% | −2.03% | −1.95% | −1.95% | −2.08% | 1.47% | 1.19% | 1.16% |
High-Low | −4.34% | −4.12% | −4.09% | −4.17% | −4.46% | 0.34% | 0.00% | −0.04% |
Panel E | ||||||||
BETA-CRSP VW Only | ||||||||
Low Beta | 1.08% | 1.11% | 1.15% | 1.16% | 1.20% | 1.29% | 1.28% | 1.27% |
2 | 1.13% | 1.17% | 1.18% | 1.20% | 1.22% | 1.37% | 1.34% | 1.34% |
3 | 1.19% | 1.18% | 1.19% | 1.26% | 1.29% | 1.42% | 1.36% | 1.40% |
4 | 1.21% | 1.24% | 1.26% | 1.27% | 1.27% | 1.37% | 1.38% | 1.31% |
5 | 1.22% | 1.27% | 1.23% | 1.29% | 1.33% | 1.42% | 1.41% | 1.44% |
6 | 1.20% | 1.21% | 1.29% | 1.32% | 1.34% | 1.44% | 1.44% | 1.37% |
7 | 1.15% | 1.14% | 1.18% | 1.24% | 1.29% | 1.40% | 1.37% | 1.37% |
8 | 1.07% | 1.12% | 1.11% | 1.13% | 1.28% | 1.29% | 1.33% | 1.32% |
9 | 1.08% | 1.11% | 1.14% | 1.20% | 1.35% | 1.25% | 1.24% | 1.18% |
High Beta | 1.28% | 1.30% | 1.33% | 1.44% | 1.54% | 1.13% | 1.21% | 1.15% |
High-Low | 0.21% | 0.19% | 0.17% | 0.28% | 0.34% | −0.16% | −0.07% | −0.13% |
Panel F | ||||||||
---|---|---|---|---|---|---|---|---|
BETA VW1-Large Low Grade Bond Weight in Market Portfolio | ||||||||
Low Beta | 1.08% | 1.12% | 1.16% | 1.18% | 1.22% | 1.31% | 1.26% | 1.26% |
2 | 1.12% | 1.19% | 1.21% | 1.23% | 1.22% | 1.36% | 1.33% | 1.33% |
3 | 1.23% | 1.19% | 1.21% | 1.20% | 1.25% | 1.36% | 1.35% | 1.30% |
4 | 1.14% | 1.17% | 1.17% | 1.23% | 1.22% | 1.34% | 1.34% | 1.31% |
5 | 1.16% | 1.20% | 1.20% | 1.24% | 1.26% | 1.41% | 1.41% | 1.37% |
6 | 1.20% | 1.22% | 1.24% | 1.27% | 1.37% | 1.40% | 1.40% | 1.39% |
7 | 1.16% | 1.16% | 1.17% | 1.27% | 1.34% | 1.39% | 1.40% | 1.37% |
8 | 1.15% | 1.14% | 1.14% | 1.18% | 1.25% | 1.33% | 1.31% | 1.31% |
9 | 1.18% | 1.23% | 1.26% | 1.24% | 1.36% | 1.30% | 1.34% | 1.27% |
High Beta | 1.21% | 1.25% | 1.29% | 1.48% | 1.61% | 1.18% | 1.23% | 1.23% |
High-Low | 0.13% | 0.13% | 0.13% | 0.30% | 0.40% | −0.14% | −0.03% | −0.02% |
without the impact of asset volatility. In the next section, I use two methods for estimating firm returns and then re-test the size, book-to-market, and default risk effects.
To calculate actual firm returns, the entire debt makeup of the firm should be considered in a weighted market value sense. Most firms have several types of debt including large portions of bank debt. Houston and James examine the debt mix of a random sample of 250 firms [
Carey studies the credit risk of private debt. Monitoring and priority play a role in the loss rates and returns of private debt [
When calculating firm returns, given the difficulty in obtaining data on firm level private and bank loan returns, I assume all debt is public with their returns equaling the returns of corporate bonds11. Since private debt is less risky, this assumption will likely bias the results for finding a size and book-to-market effect in firm returns. In addition, I do not have data on the percentage of private debt a firm holds. Estimating the ratio of private to public debt for individual firms would increase estimation error in debt returns and hence firm returns.
Choi attempts to directly measure firm returns by constructing a weighted average of equity, public bond, and bank loan returns [
I use the observed corporate bond returns from Reuters EJV to fit a model, then estimate bond returns from that model. This does not impose an equilibrium model on the data, which may or may not hold empirically, and allows for maximum retention of the universe of firms. In addition, although the fitted bond returns from a regression are an expected value, the expected value can be negative given the design of the model.
I use the following regression to fit the model to estimate bond returns:
where
The structure of the model is designed to pick up the maximum amount of cross-sectional variability in returns based on market-wide and firm-specific movements. The set of X variables includes the risk-free rate and market-wide returns. Z includes concurrent and lagged firm level stock returns. Stock returns are intended to pick up firm-specific movements in cash flow or expected return news related to bond returns.
I pool the data and assume the regression coefficients are functions of firm-specific variables in V, similar to Shanken [
I include the variables that are independent variables in the subsequent Fama-Mac- Beth regressions. This ensures that the noise induced from using fitted bond returns in the Fama-MacBeth regressions, as opposed to actual bond returns, is orthogonal to the independent variables in the regressions. Since the intercept is a function of V, the residuals are orthogonal to each component of V.
Coefficient | Standard error | t | |
---|---|---|---|
Pooled Model | |||
Intercept | 0.0123 | 0.0027 | 4.57 |
ln(ME) | −0.0008 | 0.0003 | −2.25 |
ln(BE/ME) | −0.0010 | 0.0009 | −1.12 |
ln(EDF) | 0.0015 | 0.0005 | 3.11 |
Stock Return | −0.0160 | 0.0285 | −0.56 |
Lag Stock Return | 0.0614 | 0.0165 | 3.72 |
Risk Free Rate | −0.5495 | 1.4349 | −0.38 |
High Grade Corporate Bond Return | −0.4049 | 0.1664 | −2.43 |
Low Grade Corporate Bond Return | 0.1623 | 0.0682 | 2.38 |
Value -Weighted CRSP Stock Return | 0.0663 | 0.0393 | 1.69 |
10 Year CRSP Government Bond Return | 0.0163 | 0.1271 | 0.13 |
1 Year CRSP Government Bond Return | 0.9205 | 0.5981 | 1.54 |
Interaction Variables | |||
ln(ME)*HG | 0.0246 | 0.0288 | 0.85 |
ln(ME)*LG | 0.0090 | 0.0129 | 0.7 |
ln(ME)*stock return | 0.0145 | 0.0037 | 3.89 |
ln(ME)*lag stock return | 0.0020 | 0.0018 | 1.14 |
ln(ME)*rf | −0.3549 | 0.2645 | −1.34 |
ln(ME)*VW CRSP | −0.0091 | 0.0065 | −1.4 |
ln(ME) × 10 year | −0.0706 | 0.0195 | −3.63 |
ln(ME) × 1 year | −0.0940 | 0.0977 | −0.96 |
ln(BE/ME)*HG | 0.0613 | 0.0595 | 1.03 |
ln(BE/ME)*LG | 0.1153 | 0.0259 | 4.44 |
ln(BE/ME)*stock return | −0.0017 | 0.0065 | −0.26 |
ln(BE/ME)*lag stock return | −0.0074 | 0.0035 | −2.1 |
ln(BE/ME)*rf | 0.0628 | 0.5326 | 0.12 |
ln(BE/ME)*VW CRSP | −0.0573 | 0.0142 | −4.02 |
ln(BE/ME) × 10 year | −0.0262 | 0.0429 | −0.61 |
ln(BE/ME) × 1 year | 0.0855 | 0.1750 | 0.49 |
---|---|---|---|
ln(EDF)*HG | 0.0692 | 0.0239 | 2.9 |
ln(EDF)*LG | 0.0015 | 0.0095 | 0.15 |
ln(EDF)*stock return | 0.0139 | 0.0046 | 3.06 |
ln(EDF)*lag stock return | −0.0069 | 0.0024 | −2.88 |
ln(EDF)*rf | −0.0588 | 0.1959 | −0.3 |
ln(EDF)*VW CRSP | −0.0174 | 0.0056 | −3.13 |
ln(EDF) × 10 year | 0.0102 | 0.0182 | 0.56 |
ln(EDF) × 1 year | −0.0984 | 0.0835 | −1.18 |
Average of Firm Specific Models | |||
ln(ME) | −0.0076 | 0.1534 | −1.65 |
ln(BE/ME) | −0.0036 | 0.1280 | −0.94 |
ln(EDF) | −0.0021 | 0.0387 | −1.82 |
Stock Return | 0.0559 | 0.1199 | 15.6 |
Lag Stock Return | 0.0083 | 0.0631 | 4.38 |
Risk Free Rate | 0.1648 | 22.2947 | 0.25 |
High Grade Corporate Bond Return | 0.0961 | 2.6584 | 1.21 |
Low Grade Corporate Bond Return | 0.0950 | 0.6277 | 5.06 |
Value -Weighted CRSP Stock Return | 0.0062 | 0.3019 | 0.68 |
10 Year CRSP Government Bond Return | 0.1386 | 1.5933 | 2.91 |
1 Year CRSP Government Bond Return | −0.0100 | 6.2605 | −0.05 |
sion is 10%, however the mean and median R2 for the firm-specific regressions are both above 50%. I assume that
Since the indices are correlated and there are three sets of interaction variables, the coefficients are difficult to interpret. The idea of the pooled regression is not to over-fit the model, but to explain as much of the bond returns as possible. Looking at the average coefficients for the firm-specific regressions helps to understand the economic relationship of the variables with the bond returns. The individual bond returns are strongly related to concurrent stock returns. The bond returns are also correlated with one month lagged returns, but the relationship is not as strong. Both the investment grade and speculative grade indices are positively related to bond returns with a similar average coefficient. The standard error of the average coefficients is lower for the speculative grade index. The largest average coefficient on the indices is on the 10-year index. Size, book-to-market, and EDF are all negatively related to bond returns.
To mitigate concerns about the econometric model for debt returns, I also use a simple approach to estimate bond returns using index returns of corporate bonds. While this method does not allow for firm-specific variation in the estimated debt returns, it helps gauge how important the debt return estimate is relative to controlling for the leverage impact on returns.
For each month, I estimate the median EDF for each rating. Then I interpolate between the ratings buckets to create an EDF minimum and maximum for each rating class in each month. I then assign each firm an index return for the rating group that the EDF falls in. I use the S&P ratings available on COMPUSTAT. Since the ratings are not available before 1985, I used the last year’s average cutoff values for the periods prior to 1985. I refer to this method in the results as the index method for firm returns.
Using the bond return model and the index approach, I estimate monthly debt returns for each firm and form firm returns. The weight on debt returns is the book debt definition of leverage used throughout the paper, and the weight on equity returns is one minus the leverage measure. I retain a large unbiased cross-section and time-series of monthly firm returns using this method. I first re-examine the single variable sorts and compare firm and stock returns to each other.
If the size and book-to-market premiums are positive in firm returns, it supports the notion that the premiums are derived from economic risk factors unrelated to leverage. However, if the size and book-to-market premiums are non-existent in firm returns, it suggests differences in capital structure are driving the size and book-to-market premiums. One important thing to note is that the risk premium for equity returns is the firm return grossed up by leverage. Therefore, if size and book-to-market are asset risk proxies, the premium in firm returns will be less than in equity returns. Depending on the relationship between asset risk and leverage, the coefficients on size and book-to- market may automatically decrease12. However, the t-statistics in firm return regressions do not automatically decrease. There still should be a statistically positive premium on the variables if they proxy for asset risk.
The difference between monthly stock returns of the high and low book-to-market firms is 75 basis points and is 3.4 standard errors from zero. The two sets of firm returns are similar across the book-to-market deciles. The impact of leverage starts to show a significant reduction in average returns as book-to-market decreases. The average firm returns difference between the high and low book-to-market portfolios is
VAR | TYPE | DEC10 - DEC1 | DECILE1 | DECILE2 | DECILE3 | DECILE4 | DECILE5 | DECILE6 | DECILE7 | DECILE8 | DECILE9 | DECILE10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
LN(BE\ME) | LEVEL | 3.14 | −2.52 | −1.52 | −1.15 | −0.89 | −0.68 | −0.50 | −0.32 | −0.14 | 0.08 | 0.62 |
Stock Return | Average | 0.75% | 0.91% | 0.99% | 0.98% | 1.09% | 1.04% | 1.12% | 1.25% | 1.33% | 1.39% | 1.65% |
Stock Return | SE | 0.22% | 0.31% | 0.29% | 0.27% | 0.26% | 0.25% | 0.24% | 0.24% | 0.24% | 0.25% | 0.27% |
Stock Return | T | 3.36 | 2.95 | 3.44 | 3.62 | 4.21 | 4.15 | 4.59 | 5.26 | 5.44 | 5.60 | 6.14 |
Firm Return Model | Average | 0.38% | 0.89% | 0.97% | 0.96% | 1.04% | 0.99% | 1.05% | 1.11% | 1.16% | 1.20% | 1.27% |
Firm Return Model | SE | 0.21% | 0.28% | 0.26% | 0.23% | 0.21% | 0.19% | 0.18% | 0.17% | 0.16% | 0.16% | 0.14% |
Firm Return Model | T | 1.78 | 3.13 | 3.76 | 4.14 | 4.94 | 5.13 | 5.81 | 6.66 | 7.06 | 7.59 | 9.15 |
Firm Return Index | Average | 0.30% | 0.87% | 0.95% | 0.93% | 1.01% | 0.96% | 1.00% | 1.06% | 1.10% | 1.13% | 1.17% |
Firm Return Index | SE | 0.21% | 0.29% | 0.26% | 0.24% | 0.22% | 0.20% | 0.19% | 0.18% | 0.18% | 0.18% | 0.17% |
Firm Return Index | T | 1.46 | 3.04 | 3.64 | 3.94 | 4.64 | 4.71 | 5.25 | 5.89 | 6.10 | 6.41 | 7.03 |
LN(ME) | LEVEL | 5.79 | 3.37 | 4.54 | 5.10 | 5.56 | 5.98 | 6.40 | 6.83 | 7.33 | 7.95 | 9.16 |
Stock Return | Average | −0.34% | 1.28% | 1.20% | 1.16% | 1.20% | 1.14% | 1.04% | 1.17% | 1.09% | 1.06% | 0.94% |
Stock Return | SE | 0.18% | 0.26% | 0.29% | 0.28% | 0.28% | 0.27% | 0.25% | 0.24% | 0.23% | 0.22% | 0.20% |
Stock Return | T | −1.90 | 4.93 | 4.19 | 4.07 | 4.31 | 4.30 | 4.22 | 4.81 | 4.65 | 4.81 | 4.61 |
Firm Return Model | Average | −0.33% | 1.17% | 1.08% | 1.07% | 1.09% | 1.03% | 0.92% | 1.06% | 0.97% | 0.93% | 0.84% |
Firm Return Model | SE | 0.13% | 0.19% | 0.22% | 0.22% | 0.21% | 0.20% | 0.19% | 0.19% | 0.18% | 0.17% | 0.16% |
Firm Return Model | T | −2.46 | 6.25 | 4.89 | 4.89 | 5.09 | 5.01 | 4.88 | 5.67 | 5.48 | 5.58 | 5.19 |
Firm Return Index | Average | −0.21% | 1.08% | 1.03% | 1.02% | 1.05% | 0.99% | 0.89% | 1.04% | 0.96% | 0.93% | 0.87% |
Firm Return Index | SE | 0.13% | 0.20% | 0.23% | 0.23% | 0.22% | 0.21% | 0.20% | 0.20% | 0.19% | 0.17% | 0.17% |
Firm Return Index | T | −1.57 | 5.38 | 4.43 | 4.47 | 4.69 | 4.62 | 4.52 | 5.30 | 5.18 | 5.36 | 5.22 |
MKMV EDF | LEVEL | 3.80 | −2.31 | −1.50 | −1.11 | −0.79 | −0.51 | −0.22 | 0.07 | 0.39 | 0.79 | 1.49 |
Stock Return | Average | 0.05% | 1.04% | 1.08% | 1.09% | 1.27% | 1.24% | 1.20% | 1.25% | 1.28% | 1.22% | 1.09% |
Stock Return | SE | 0.22% | 0.19% | 0.21% | 0.22% | 0.24% | 0.25% | 0.26% | 0.28% | 0.30% | 0.31% | 0.34% |
Stock Return | T | 0.22 | 5.54 | 5.25 | 4.94 | 5.36 | 4.96 | 4.63 | 4.52 | 4.30 | 3.93 | 3.24 |
Firm Return Model | Average | −0.08% | 1.02% | 1.03% | 1.01% | 1.15% | 1.13% | 1.08% | 1.12% | 1.11% | 1.04% | 0.94% |
Firm Return Model | SE | 0.12% | 0.17% | 0.18% | 0.18% | 0.19% | 0.20% | 0.20% | 0.21% | 0.22% | 0.21% | 0.19% |
Firm Return Model | T | −0.67 | 6.01 | 5.90 | 5.54 | 5.97 | 5.63 | 5.32 | 5.34 | 5.09 | 4.93 | 4.81 |
Firm Return Index | Average | −0.09% | 1.00% | 1.00% | 0.98% | 1.10% | 1.08% | 1.02% | 1.05% | 1.04% | 0.99% | 0.91% |
Firm Return Index | SE | 0.13% | 0.17% | 0.18% | 0.19% | 0.20% | 0.21% | 0.21% | 0.22% | 0.23% | 0.23% | 0.22% |
Firm Return Index | T | −0.69 | 5.81 | 5.58 | 5.21 | 5.56 | 5.20 | 4.78 | 4.70 | 4.45 | 4.27 | 4.13 |
around half of the stock returns difference. The difference in monthly firm returns is 38 and 30 basis points depending on the bond return method used. The average differences in firm returns between the high and low portfolios are both less than two standard errors from zero.
The results are not as strong when examining the size returns. This is expected since book-to-market is more correlated with leverage than size. The small cap stock issue is mitigated by deleting the stocks with prices below $5.00 and by using the NYSE size breakpoints. The average return difference for stocks is −0.34% per month and the difference in firm returns is −0.33% or −0.21%, depending on the definition. Accounting for the leverage differences has a small impact on the size premium.
The results when sorting by EDF show very little difference in average returns across the EDF deciles. When accounting for the leverage impact, the difference becomes slightly negative between high and low EDF firms. These results are not consistent with the Campbell et al. results or Vassalou and Xing, but are consistent with studies that argue that the default risk results are driven by small cap firms [
The formal test of the return relationships is done using the Fama-MacBeth regression approach (
In the first specification, the average coefficients on ln(BE/ME) is 0.226% for the stock return regression and drops by more than half for both sets of firm returns. The average coefficient on ln(BE/ME) is 2.8 standard errors from zero in the stock return regression and 1.2 and 1.0 for the firm returns regressions. This is strong evidence that value premium is a capital structure effect and has little to do with asset volatility. This is further supported in the third specification where ln(EDF) is included in the regressions. The coefficients and t-statistics on ln(BE/ME) are very similar to the specification without ln(EDF).
A very different pattern emerges when focusing on the ln(ME) coefficients. The negative relationship between stock returns and size is not statistically meaningful. After controlling for leverage, the size coefficient actually becomes more negative in the first firm return definition and is unaffected in the second. However, neither of the firm returns average coefficients are more than two standard errors from zero.
The default risk premium does not exist in stock returns when included alone in specification two13. After controlling for debt returns, the EDF and firm return relationship turns negative, but is not statistically significant. In the third specification including all three variables, EDF is now negatively related to stock returns. In addition, the relationship between EDF and firm returns is negative and more than two standard errors from zero for both firm returns specifications. This implies that the correlation between
ln(BE/ME) | ln(ME) | ln(EDF) | ||
---|---|---|---|---|
Stock Return | Avg Coef. | 0.226% | −0.024% | |
Stock Return | t-stat | 2.78 | −0.64 | |
Firm Return Model | Avg Coef. | 0.099% | −0.040% | |
Firm Return Model | t-stat | 1.20 | −1.30 | |
Firm Return Index | Avg Coef. | 0.082% | −0.024% | |
Firm Return Index | t-stat | 1.04 | −0.77 | |
Stock Return | Avg Coef. | 0.022% | ||
Stock Return | t-stat | 0.45 | ||
Firm Return Model | Avg Coef. | −0.010% | ||
Firm Return Model | t-stat | −0.33 | ||
Firm Return Index | Avg Coef. | −0.015% | ||
Firm Return Index | t-stat | −0.45 | ||
Stock Return | Avg Coef. | 0.233% | −0.057% | −0.071% |
Stock Return | t-stat | 2.71 | −1.75 | −1.38 |
Firm Return Model | Avg Coef. | 0.120% | −0.081% | −0.095% |
Firm Return Model | t-stat | 1.42 | −2.78 | −3.11 |
Firm Return Index | Avg Coef. | 0.097% | −0.059% | −0.079% |
Firm Return Index | t-stat | 1.18 | −2.09 | −2.35 |
EDF and size has a strong impact on the regression outcome, since book-to-market is relatively unaffected. After controlling for debt returns, both size and EDF are negatively related to returns. Since leverage is mechanically controlled for in the firm returns, this must mean the negative part is related to asset volatility.
The results in this paper establish that the separation of the components of default risk is an important step in understanding the relationship between stock returns, book-to- market, and size. Default risk is driven by leverage and asset volatility, however only one of these is directly related to systematic risk. The results demonstrate that leverage is moving over time due to changes in market values. The dynamic nature of leverage is not captured by standard measures of equity beta. I show that multiple definitions of the market portfolio do not change this result.
By focusing on leverage only, I show that a large portion of the book-to-market premium in stock returns is explained away by capital structure difference across firms. However, I show that there is no meaningful default risk premium in stock returns or firm returns. After controlling for size and book-to-market, default risk is negatively related to firm returns. Since leverage is neutralized, asset volatility is the remaining component negatively related to firm returns.
I have contributed to the literature seeking to explain the size and value premiums in stock returns with an original approach for neutralizing risk associated with financial leverage. This is the first study comparable in length and breadth to Fama and French that looks at asset or firm returns [
The new approach used to estimated debt returns also has limitations. Since we are not able to observe the actual market value of the firm, the estimates will lead to errors. The tradeoff is sample size and estimation error. In addition, while I am able to explain a portion of the book-to-market premium, there is still a positive coefficient in firm returns. It is possible that book-to-market is correlated with leverage and other firm level risk factors.
Hood III, F.M. (2016) Leverage, Default Risk, and the Cross- Section of Equity and Firm Returns. Mo- dern Economy, 7, 1610-1639. http://dx.doi.org/10.4236/me.2016.714143