TY - JOUR
AU - Hou, Kewei
AB - Abstract We study the effect of a bond’s place in its issuer’s maturity structure on credit risk. Using a structural model as motivation, we argue that bonds due relatively late in their issuers’ maturity structure have greater credit risk than do bonds due relatively early. Empirically, we find robust evidence that these later bonds have larger yield spreads and greater comovement with equity and that the magnitude of the effects is consistent with model predictions for investment-grade bonds. Our results highlight the importance of bond-specific credit risk for understanding corporate bond prices. Received January 1, 2016; editorial decision June 6, 2017 by Editor Robin Greenwood. One of the central issues in the corporate bond literature is about measuring credit risk and its effects on prices and returns. In this paper, we will focus on understanding the empirical implications of de facto seniority, the idea that bonds due relatively late in their issuers’ maturity structure are effectively junior even if firms do not have explicit seniority structures. Much of the literature since Merton (1974) has tried to understand the levels and dynamics of corporate bond prices with a focus on more realistic modeling of firms (including bankruptcy costs, endogenizing default decisions, interest rate dynamics, and jumps). Our particular focus is on better understanding how there can be heterogeneity in credit risk for bonds based on de facto seniority, with an emphasis on the impact on bond returns and the cross-section of yield spreads. We control for firm-level credit risk, the actual maturity of a corporate bond, and the average maturity of all of the issuer’s corporate bonds, and find that bonds due relatively late in their issuers’ maturity structure have higher yield spreads and higher hedge ratios.1 Thus, we will argue that, in addition to firm-level credit risk, it is important to consider bond-level credit risk based on the place of a bond in its issuer’s maturity structure. The intuition behind our empirical results is that a bond that matures after most of the other bonds issued by the same firm is potentially de facto junior even if all of the firm’s bonds have the same explicit seniority.2 This arises from the fact that a firm in financial trouble may remain solvent long enough to repay bonds due early in its maturity structure, but not bonds due later. The effect is that bond issues that mature later are more sensitive to underlying firm value. We formalize this intuition in an extension of the Merton (1974) model and present both yield spreads and relative hedge ratios in a numerical exercise that shows that, holding the actual maturity constant, bonds due later in their firms’ maturity structure have higher yield spreads and hedge ratios. Empirically, we find that after controlling for various measures of credit risk (including credit ratings), measures of illiquidity, maturity (both of the bond and the average across the issuer’s bonds), and firm fixed effects, bonds due latest in their issuer’s maturity structure have yield spreads that are 48 basis points higher than bonds due earliest in their issuer’s maturity structure. This is larger than an estimated 21-basis-point increase in yield spreads associated with a one-notch deterioration in credit rating.3 The effect is 47 basis points for investment-grade bonds, compared to 85 basis points for junk-rated bonds. Consistent with model predictions, de facto seniority becomes generally more important as credit quality begins to decline. Our results suggest that the market recognizes the effect of a bond’s place in its issuer’s maturity structure when pricing bonds and that credit ratings agencies should consider notching bonds based on de facto seniority in addition to the explicit priority structure.4 We also test how the magnitudes of our empirical estimates match with the predictions of the extended Merton model. Although the model is largely meant to illustrate intuition and motivate the de facto seniority effect, it is nevertheless useful to understand how well the model fits and identify the types of firms that the model is best able to explain. In our tests, we find that the empirically estimated de facto seniority effect in yield spreads is significantly below the model predicted effect for junk-rated bonds. However, for most specifications, we are unable to reject that model and empirical magnitudes are equal for investment-grade firms. Thus, despite the fact that we find a stronger empirical effect for junk-rated firms, the effects are not as large as the model suggests. Our results are consistent with the fact that junk-rated firms have complex capital structures (Rauh and Sufi 2010) that are not fully incorporated in the extended Merton model. Investment-grade firms, in contrast, have much simpler debt structures, so our extension of the Merton model provides a reasonable approximation. In addition to yield spreads, we also consider hedge ratios, the relative returns of corporate bonds and equities. Intuitively, hedge ratios measure how equity-like a corporate bond is, with bonds with greater credit risk behaving in a more equity-like way. Hedge ratios provide not only an alternative setting to test for the de facto seniority effect but one that structural models have performed better on than yield spreads based on previous literature (Schaefer and Strebulaev 2008). Consistent with the predictions of the extended Merton model, we find that de facto junior bonds have larger hedge ratios than do de facto senior bonds. Hence, our results using hedge ratios confirm our yield spreads results. Our paper is primarily related to two strands of literature. Empirically, our paper is related to the literature that tries to evaluate the pricing of corporate bonds through the lens of structural models of default. Huang and Huang (2003) is the seminal paper showing that a large part of credit spreads cannot be explained by common structural models of default.5 Our paper is also related to the theoretical literature on equilibrium maturity choice, particularly Brunnermeier and Oehmke (2013),6 who show that the incentive to shorten the maturity structure in conjunction with an inability to commit to a maturity structure leads to a maturity rat race for financial firms. The outcome of this maturity rat race is inefficiently short-term financing. In this paper, we do not aim to explain why a firm has chosen a particular maturity structure; instead, we take the outcome of maturity choice as given and focus on the empirical effects of a bond’s position in its firm’s maturity structure on its prices and returns. Corporate bond issuers will have both de facto senior and de facto junior bonds regardless of the equilibrium choice to largely issue short-term versus long-term bonds as de facto seniority is defined relative to bonds from the same issuer, not based on the absolute maturity of the bond. Importantly, we note that our results are robust to controlling for the average firm-level maturity (the outcome of the endogenous choices discussed in the aforementioned papers) and also the actual maturity of the bond. 1. De Facto Seniority and Credit Risk As a simple benchmark model, we consider a Merton model in which firm value follows a geometric Brownian motion under the risk-neutral probability measure \begin{align} d\ln V_t = \left(r-\frac{1}{2}\sigma_v^2\right)dt + \sigma_v dW_t^Q. \end{align}(1) The firm has a single zero-coupon bond issue with face value |$K$| and equity is a call option on the firm. The single bond issue is equivalent to a risk-free bond short a put option on the firm. Its value is \begin{align}\label{eq_merton_bond} B &= V\left(1-N(d_1)\right) + Ke^{-rT}N\left(d_2\right)\text{,}\\\notag \text{where } d_1 &= \frac{\ln\left(\frac{V}{K}\right) + \left(r + \frac{1}{2}\sigma_v^2\right)T}{\sigma_v\sqrt{T}}\text{and} d_2 = d_1 - \sigma_v\sqrt{T}\text{.} \end{align}(2) Bond yields can be directly inferred from |$B = K e^{-yT}$|⁠, and the yield spread equals |$y-r$|⁠. In addition to yield spreads, we also consider hedge ratios, the relative returns of corporate bonds and equities. As illustrated by Schaefer and Strebulaev (2008), the relative returns of the bond and equity (the hedge ratio) under the Merton model is \begin{align}\label{eq_merton_hedge} &h_E \equiv \frac{\partial \ln B}{\partial \ln E} = \left(\frac{1}{N(d_1)} - 1\right)\left(\frac{V}{B} -1\right). \end{align}(3) The Merton model formalizes the important insight that corporate bonds and equities are linked through their exposures to the underlying firm value.7 Intuitively, safe corporate bonds will be similar to Treasuries and have low hedge ratios. Corporate bonds with significant credit risk will have higher exposures to underlying firm values and will tend to have larger hedge ratios. We extend the Merton model to analyze how the position of a bond in its issuer’s maturity structure affects its yield spread and hedge ratio. The intuition behind why a bond’s position affects its credit risk is that a firm in financial distress may remain solvent long enough to repay bond issues that mature early. However, the firm is then likely to suffer solvency issues when the bonds due later eventually mature. This creates a de facto seniority effect.8 Our extension of the Merton model includes three zero-coupon bond issues, which have the same explicit seniority, but mature at different times. The three bond issues have face values |$K_i$| and maturities |$T_i$| where |$T_1 < T_2 < T_3$|⁠. At |$T_i$|⁠, a firm remains solvent if |$V_i > K_i$|⁠.9 Otherwise the firm defaults, |$V_i$| is discounted by a proportional bankruptcy cost (⁠|$L$|⁠), and the remaining bond issues share the remaining firm value in proportion to their face values.10|$^,$|11 This model, although a simplification of the complex capital structure typical in large U.S. corporations, allows us to examine the yield spread and hedge ratio, while varying the bond’s position in the firm’s maturity structure. By including three bonds, we are able to focus on the bond with the intermediate maturity, |$T_2$|⁠, and adjust the relative amounts of debt due before and after it.12 This changes the de facto seniority of the intermediate maturity bond, holding all else constant, allowing us to gauge the impact of de facto seniority on yield spreads and hedge ratios. In this extended Merton model, equity remains a call option on the firm, but has intermediate monitoring points at each bond maturity \begin{align} E = E_0^Q\left[e^{-rT_3}{\rm 1}\kern-0.24em{\rm I}_{V_1 > K_1}{\rm 1}\kern-0.24em{\rm I}_{V_2>K_2}{\rm 1}\kern-0.24em{\rm I}_{V_3>K_3}\left(V_3 - K_3\right)\right]\text{.} \end{align}(4) Though there is no closed-form solution for equity value, it can be simplified to a function of normal cumulative distribution functions (CDFs) and integrals by noting that firm value returns are lognormal. Similarly, the value of the bond maturing at |$T_2$| is \begin{align}\label{eq_int_bond} B &= E_0^Q\left[e^{-rT_2}{\rm 1}\kern-0.24em{\rm I}_{V_1 > K_1}{\rm 1}\kern-0.24em{\rm I}_{V_2>K_2}K_2\right] + E_0^Q\left[e^{-rT_1}{\rm 1}\kern-0.24em{\rm I}_{V_1 < K_1}(1-L)V_1\frac{K_2}{K_1+K_2+K_3}\right]\nonumber\\ &\quad + E_0^Q\left[e^{-rT_2}{\rm 1}\kern-0.24em{\rm I}_{V_1 > K_1}{\rm 1}\kern-0.24em{\rm I}_{V_2<K_2}(1-L)V_2\frac{K_2}{K_2 + K_3}\right], \end{align}(5) where the three terms represent solvency at |$T_2$|⁠, default at |$T_1$|⁠, and default at |$T_2$|⁠, respectively. The yield spread can then be calculated from bond prices and the hedge ratio |$\left(\frac{\partial \ln B}{\partial \ln E}\right)$| can be calculated using numerical procedures. Appendix A provides details for the pricing equations. In Table 1, we consider the yield spreads for corporate bonds under both the Merton model and the extended Merton model for various leverage-asset volatility combinations.13 In panel A, we focus on the Merton model as a benchmark, finding the expected result that as credit risk increases (leverage and asset volatility increase), yield spreads increase. This can also be seen in Figure 1(A). We note that the model yield spreads are somewhat lower than typical empirical yield spreads, consistent with Eom, Helwege, and Huang (2004) and Huang and Huang (2012), who show that Merton-type models typically underpredict yield spreads. Figure 1 Open in new tabDownload slide Yield spreads and hedge ratios for the Merton and extended Merton models. The cyan surfaces represent the case in which a bond is late in its firm’s maturity structure and the red surfaces with point markers represent the case in which a bond is early in its firm’s maturity structure (B and D). Table 1 Model yield spreads A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0033 0.1795 1.8191 23.3688 95.0277 0.2 0.0000 0.0070 0.5421 5.0495 19.6809 99.2050 262.8057 0.3 0.0004 0.3458 5.3895 23.9262 62.5342 211.1534 460.5633 Market leverage 0.4 0.0401 3.2000 21.0989 62.9217 131.9890 353.8877 685.6167 0.5 0.7336 13.6751 54.1020 126.3897 230.3939 529.3418 943.3421 0.6 5.1229 39.0175 111.2248 220.3074 363.7037 745.5550 1,245.5468 0.7 20.7393 89.1734 202.8052 356.0352 544.9413 1,020.0144 1,614.9851 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0003 0.2851 6.9545 0.0000 0.0000 0.0002 0.0370 0.7223 15.6811 75.7137 0.2 0.0000 0.0000 0.0000 0.0003 0.0296 3.2024 31.0985 0.0000 0.0005 0.1928 3.2304 16.0604 94.8426 259.9769 0.3 0.0000 0.0000 0.0000 0.0074 0.2631 10.3957 65.2067 0.0000 0.1184 4.1320 23.6731 67.4071 234.7669 505.9163 Market leverage 0.4 0.0000 0.0000 0.0003 0.0521 0.9820 21.1628 102.7301 0.0053 2.4346 23.6895 77.5608 165.0325 430.6490 802.983 0.5 0.0000 0.0000 0.0022 0.1861 2.3027 33.6428 138.9410 0.3388 16.2183 75.1135 177.4995 316.5454 683.2978 1,155.9533 0.6 0.0000 0.0000 0.0084 0.4244 4.0014 45.6987 170.4882 5.0106 59.6523 175.7486 337.2298 533.5227 1,007.8228 1,571.5186 0.7 0.0000 0.0000 0.0188 0.6912 5.5846 55.4812 196.0989 31.2864 158.1604 348.5061 578.4603 836.3610 1,414.3348 2,063.3007 A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0033 0.1795 1.8191 23.3688 95.0277 0.2 0.0000 0.0070 0.5421 5.0495 19.6809 99.2050 262.8057 0.3 0.0004 0.3458 5.3895 23.9262 62.5342 211.1534 460.5633 Market leverage 0.4 0.0401 3.2000 21.0989 62.9217 131.9890 353.8877 685.6167 0.5 0.7336 13.6751 54.1020 126.3897 230.3939 529.3418 943.3421 0.6 5.1229 39.0175 111.2248 220.3074 363.7037 745.5550 1,245.5468 0.7 20.7393 89.1734 202.8052 356.0352 544.9413 1,020.0144 1,614.9851 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0003 0.2851 6.9545 0.0000 0.0000 0.0002 0.0370 0.7223 15.6811 75.7137 0.2 0.0000 0.0000 0.0000 0.0003 0.0296 3.2024 31.0985 0.0000 0.0005 0.1928 3.2304 16.0604 94.8426 259.9769 0.3 0.0000 0.0000 0.0000 0.0074 0.2631 10.3957 65.2067 0.0000 0.1184 4.1320 23.6731 67.4071 234.7669 505.9163 Market leverage 0.4 0.0000 0.0000 0.0003 0.0521 0.9820 21.1628 102.7301 0.0053 2.4346 23.6895 77.5608 165.0325 430.6490 802.983 0.5 0.0000 0.0000 0.0022 0.1861 2.3027 33.6428 138.9410 0.3388 16.2183 75.1135 177.4995 316.5454 683.2978 1,155.9533 0.6 0.0000 0.0000 0.0084 0.4244 4.0014 45.6987 170.4882 5.0106 59.6523 175.7486 337.2298 533.5227 1,007.8228 1,571.5186 0.7 0.0000 0.0000 0.0188 0.6912 5.5846 55.4812 196.0989 31.2864 158.1604 348.5061 578.4603 836.3610 1,414.3348 2,063.3007 Panel A reports Merton model yield spreads, where the bond price follows from Equation (2) and yields are calculated from the bond price before the risk-free rate (3%) is subtracted. Panel B reports Merton model yield spreads for the intermediate maturity bond priced in Equation (5). A firm is assumed to have three zero-coupon bonds with maturities |$T_1 = 5$|⁠, |$T_2 = 7$|⁠, and |$T_3 = 10$|⁠. For each market leverage-asset volatility combination, the face values of the three bond issues, |$K_1$|⁠, |$K_2$|⁠, and |$K_3$|⁠, are chosen to match the market leverage. The top number for each market leverage-asset volatility combination corresponds to 10% of the market value being attributable to the short maturity bond, 10% to the intermediate maturity bond, and 80% to the long maturity bond. In this case, most of the firm’s debt is due after the intermediate bond, so the intermediate bond is de facto senior. The bottom number corresponds to 80% attributable to the short maturity bond, 10% to the intermediate maturity bond, and 10% to the long maturity bond. In this case, the intermediate bond is de facto junior. Yield spreads are reported in basis points. Open in new tab Table 1 Model yield spreads A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0033 0.1795 1.8191 23.3688 95.0277 0.2 0.0000 0.0070 0.5421 5.0495 19.6809 99.2050 262.8057 0.3 0.0004 0.3458 5.3895 23.9262 62.5342 211.1534 460.5633 Market leverage 0.4 0.0401 3.2000 21.0989 62.9217 131.9890 353.8877 685.6167 0.5 0.7336 13.6751 54.1020 126.3897 230.3939 529.3418 943.3421 0.6 5.1229 39.0175 111.2248 220.3074 363.7037 745.5550 1,245.5468 0.7 20.7393 89.1734 202.8052 356.0352 544.9413 1,020.0144 1,614.9851 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0003 0.2851 6.9545 0.0000 0.0000 0.0002 0.0370 0.7223 15.6811 75.7137 0.2 0.0000 0.0000 0.0000 0.0003 0.0296 3.2024 31.0985 0.0000 0.0005 0.1928 3.2304 16.0604 94.8426 259.9769 0.3 0.0000 0.0000 0.0000 0.0074 0.2631 10.3957 65.2067 0.0000 0.1184 4.1320 23.6731 67.4071 234.7669 505.9163 Market leverage 0.4 0.0000 0.0000 0.0003 0.0521 0.9820 21.1628 102.7301 0.0053 2.4346 23.6895 77.5608 165.0325 430.6490 802.983 0.5 0.0000 0.0000 0.0022 0.1861 2.3027 33.6428 138.9410 0.3388 16.2183 75.1135 177.4995 316.5454 683.2978 1,155.9533 0.6 0.0000 0.0000 0.0084 0.4244 4.0014 45.6987 170.4882 5.0106 59.6523 175.7486 337.2298 533.5227 1,007.8228 1,571.5186 0.7 0.0000 0.0000 0.0188 0.6912 5.5846 55.4812 196.0989 31.2864 158.1604 348.5061 578.4603 836.3610 1,414.3348 2,063.3007 A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0033 0.1795 1.8191 23.3688 95.0277 0.2 0.0000 0.0070 0.5421 5.0495 19.6809 99.2050 262.8057 0.3 0.0004 0.3458 5.3895 23.9262 62.5342 211.1534 460.5633 Market leverage 0.4 0.0401 3.2000 21.0989 62.9217 131.9890 353.8877 685.6167 0.5 0.7336 13.6751 54.1020 126.3897 230.3939 529.3418 943.3421 0.6 5.1229 39.0175 111.2248 220.3074 363.7037 745.5550 1,245.5468 0.7 20.7393 89.1734 202.8052 356.0352 544.9413 1,020.0144 1,614.9851 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0003 0.2851 6.9545 0.0000 0.0000 0.0002 0.0370 0.7223 15.6811 75.7137 0.2 0.0000 0.0000 0.0000 0.0003 0.0296 3.2024 31.0985 0.0000 0.0005 0.1928 3.2304 16.0604 94.8426 259.9769 0.3 0.0000 0.0000 0.0000 0.0074 0.2631 10.3957 65.2067 0.0000 0.1184 4.1320 23.6731 67.4071 234.7669 505.9163 Market leverage 0.4 0.0000 0.0000 0.0003 0.0521 0.9820 21.1628 102.7301 0.0053 2.4346 23.6895 77.5608 165.0325 430.6490 802.983 0.5 0.0000 0.0000 0.0022 0.1861 2.3027 33.6428 138.9410 0.3388 16.2183 75.1135 177.4995 316.5454 683.2978 1,155.9533 0.6 0.0000 0.0000 0.0084 0.4244 4.0014 45.6987 170.4882 5.0106 59.6523 175.7486 337.2298 533.5227 1,007.8228 1,571.5186 0.7 0.0000 0.0000 0.0188 0.6912 5.5846 55.4812 196.0989 31.2864 158.1604 348.5061 578.4603 836.3610 1,414.3348 2,063.3007 Panel A reports Merton model yield spreads, where the bond price follows from Equation (2) and yields are calculated from the bond price before the risk-free rate (3%) is subtracted. Panel B reports Merton model yield spreads for the intermediate maturity bond priced in Equation (5). A firm is assumed to have three zero-coupon bonds with maturities |$T_1 = 5$|⁠, |$T_2 = 7$|⁠, and |$T_3 = 10$|⁠. For each market leverage-asset volatility combination, the face values of the three bond issues, |$K_1$|⁠, |$K_2$|⁠, and |$K_3$|⁠, are chosen to match the market leverage. The top number for each market leverage-asset volatility combination corresponds to 10% of the market value being attributable to the short maturity bond, 10% to the intermediate maturity bond, and 80% to the long maturity bond. In this case, most of the firm’s debt is due after the intermediate bond, so the intermediate bond is de facto senior. The bottom number corresponds to 80% attributable to the short maturity bond, 10% to the intermediate maturity bond, and 10% to the long maturity bond. In this case, the intermediate bond is de facto junior. Yield spreads are reported in basis points. Open in new tab More importantly, in panel B of Table 1, we examine the yield spreads of bonds that are de facto senior and de facto junior. We consider two scenarios. In both scenarios, we fix the value of the bond due at |$T_2$| to be 10% of the total market value of debt. In the first scenario, 80% of the firm’s market value of debt is from the bond due at |$T_3$| and only 10% of the market value of debt is from the bond due at |$T_1$|⁠. The intermediate maturity bond, due at |$T_2$|⁠, is de facto senior to most of the debt in the firm. In the second scenario, we flip the proportions of the market value of debt due to the bonds maturing at |$T_1$| and |$T_3$|⁠, making the intermediate maturity bond de facto junior to most of the firm’s debt.14 We set the maturities of the three bonds to be |$T_1 = 5$|⁠, |$T_2 = 7$|⁠, and |$T_3 = 10$|⁠, and the bankruptcy cost to be |$L = 0.5$|⁠. Panel B shows that, at extremely low levels of credit risk, whether a bond is de facto senior or de facto junior has little effect on the yield spread. At more moderate levels, we begin to see that de facto senior bonds continue to have very small yield spreads, whereas de facto junior bonds have higher yield spreads. For example, at an asset volatility of 20% and a leverage of 50%, the de facto senior bond still has a yield spread close to zero, whereas the de facto junior bond has a yield spread of 75 basis points. As a benchmark, in the basic Merton model, the single bond has a yield spread of 54 basis points. At high levels of credit risk, de facto seniority becomes even more important. For example, at an asset volatility of 40% and a leverage of 60%, the model yield spread is only 46 basis points for the de facto senior bond as compared to 10% for the de facto junior bond. In Table 2 and Figures 1(c) and 1(d), we study hedge ratios. Hedge ratios measure the relative returns of corporate bonds and equities, with higher hedge ratios representing cases where a bond is more equity-like. Our numerical results for hedge ratios bear out a similar story to yield spreads. As credit quality of a firm declines, hedge ratios increase, and much more so for the de facto junior bond. For example, under the Merton model, a firm with an asset volatility of 10% and a leverage of 20% has a hedge ratio of virtually zero under the Merton model. Under the extended Merton model, bonds issued by such a firm have a hedge ratio close to zero regardless of whether the bonds are de facto senior or de facto junior. For an asset volatility of 20% and a leverage of 50%, the hedge ratio for the Merton model is 7.1%. That is, a 10% equity return corresponds to a 0.71% corporate bond return. For the extended Merton model, we see that de facto seniority is important for such a firm. The de facto senior bond has a virtually zero hedge ratio. The de facto junior bond has a hedge ratio of 11.9%, so a 10% equity return corresponds to a 1.19% corporate bond return. Finally, for an asset volatility of 40% and a leverage of 60%, the de facto senior bond has a hedge ratio of 4.3%, whereas the de facto junior bond has a much larger hedge ratio at 38.4%. Table 2 Model hedge ratios A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0018 0.0621 0.4412 3.2324 8.4878 0.2 0.0000 0.0042 0.1898 1.1643 3.2411 9.6727 17.1089 0.3 0.0003 0.1437 1.3279 3.9790 7.5999 15.7851 23.6740 Market leverage 0.4 0.0243 0.9330 3.7700 7.8166 12.2781 21.0071 28.7355 0.5 0.2987 2.8144 7.1155 11.9300 16.6855 25.3360 32.6755 0.6 1.3750 5.6615 10.8029 15.8607 20.5773 28.8424 35.7118 0.7 3.5755 9.0054 14.3865 19.3400 23.8277 31.5562 37.9315 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0002 0.0695 1.0461 0.0000 0.0000 0.0001 0.0174 0.2339 2.7979 8.4882 0.2 0.0000 0.0000 0.0000 0.0001 0.0099 0.5905 3.5871 0.0000 0.0004 0.0917 0.9799 3.3836 11.3312 20.1151 0.3 0.0000 0.0000 0.0000 0.0028 0.0696 1.5494 6.1945 0.0000 0.0679 1.3394 4.9732 10.0149 20.4587 29.3097 Market leverage 0.4 0.0000 0.0000 0.0001 0.0159 0.2121 2.6408 8.3368 0.0047 0.9530 5.3511 11.6261 17.8715 28.2927 36.0544 0.5 0.0000 0.0000 0.0008 0.0467 0.4153 3.6005 9.8621 0.1952 4.3125 11.9132 19.2059 25.2896 34.3312 40.6166 0.6 0.0000 0.0000 0.0026 0.0888 0.6138 4.2703 10.7585 1.8473 10.6608 19.5015 26.2566 31.3300 38.3678 43.1563 0.7 0.0000 0.0000 0.0048 0.1226 0.7392 4.5833 11.0669 7.0904 18.6201 26.5155 31.6959 35.2889 40.1597 43.7222 A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0018 0.0621 0.4412 3.2324 8.4878 0.2 0.0000 0.0042 0.1898 1.1643 3.2411 9.6727 17.1089 0.3 0.0003 0.1437 1.3279 3.9790 7.5999 15.7851 23.6740 Market leverage 0.4 0.0243 0.9330 3.7700 7.8166 12.2781 21.0071 28.7355 0.5 0.2987 2.8144 7.1155 11.9300 16.6855 25.3360 32.6755 0.6 1.3750 5.6615 10.8029 15.8607 20.5773 28.8424 35.7118 0.7 3.5755 9.0054 14.3865 19.3400 23.8277 31.5562 37.9315 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0002 0.0695 1.0461 0.0000 0.0000 0.0001 0.0174 0.2339 2.7979 8.4882 0.2 0.0000 0.0000 0.0000 0.0001 0.0099 0.5905 3.5871 0.0000 0.0004 0.0917 0.9799 3.3836 11.3312 20.1151 0.3 0.0000 0.0000 0.0000 0.0028 0.0696 1.5494 6.1945 0.0000 0.0679 1.3394 4.9732 10.0149 20.4587 29.3097 Market leverage 0.4 0.0000 0.0000 0.0001 0.0159 0.2121 2.6408 8.3368 0.0047 0.9530 5.3511 11.6261 17.8715 28.2927 36.0544 0.5 0.0000 0.0000 0.0008 0.0467 0.4153 3.6005 9.8621 0.1952 4.3125 11.9132 19.2059 25.2896 34.3312 40.6166 0.6 0.0000 0.0000 0.0026 0.0888 0.6138 4.2703 10.7585 1.8473 10.6608 19.5015 26.2566 31.3300 38.3678 43.1563 0.7 0.0000 0.0000 0.0048 0.1226 0.7392 4.5833 11.0669 7.0904 18.6201 26.5155 31.6959 35.2889 40.1597 43.7222 Panel A reports hedge ratios for the Merton model, |$\frac{\partial \ln B}{\partial \ln E}$|⁠. The Merton hedge ratio follows from Equation (3). Panel B reports extended Merton model hedge ratios. Parameters are from Table 1. Hedge ratios are reported in percentage. Open in new tab Table 2 Model hedge ratios A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0018 0.0621 0.4412 3.2324 8.4878 0.2 0.0000 0.0042 0.1898 1.1643 3.2411 9.6727 17.1089 0.3 0.0003 0.1437 1.3279 3.9790 7.5999 15.7851 23.6740 Market leverage 0.4 0.0243 0.9330 3.7700 7.8166 12.2781 21.0071 28.7355 0.5 0.2987 2.8144 7.1155 11.9300 16.6855 25.3360 32.6755 0.6 1.3750 5.6615 10.8029 15.8607 20.5773 28.8424 35.7118 0.7 3.5755 9.0054 14.3865 19.3400 23.8277 31.5562 37.9315 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0002 0.0695 1.0461 0.0000 0.0000 0.0001 0.0174 0.2339 2.7979 8.4882 0.2 0.0000 0.0000 0.0000 0.0001 0.0099 0.5905 3.5871 0.0000 0.0004 0.0917 0.9799 3.3836 11.3312 20.1151 0.3 0.0000 0.0000 0.0000 0.0028 0.0696 1.5494 6.1945 0.0000 0.0679 1.3394 4.9732 10.0149 20.4587 29.3097 Market leverage 0.4 0.0000 0.0000 0.0001 0.0159 0.2121 2.6408 8.3368 0.0047 0.9530 5.3511 11.6261 17.8715 28.2927 36.0544 0.5 0.0000 0.0000 0.0008 0.0467 0.4153 3.6005 9.8621 0.1952 4.3125 11.9132 19.2059 25.2896 34.3312 40.6166 0.6 0.0000 0.0000 0.0026 0.0888 0.6138 4.2703 10.7585 1.8473 10.6608 19.5015 26.2566 31.3300 38.3678 43.1563 0.7 0.0000 0.0000 0.0048 0.1226 0.7392 4.5833 11.0669 7.0904 18.6201 26.5155 31.6959 35.2889 40.1597 43.7222 A. Merton model . . . |$\sigma_v$| . . . 0.1 . 0.15 . 0.2 . 0.25 . 0.3 . 0.4 . 0.5 . 0.1 0.0000 0.0000 0.0018 0.0621 0.4412 3.2324 8.4878 0.2 0.0000 0.0042 0.1898 1.1643 3.2411 9.6727 17.1089 0.3 0.0003 0.1437 1.3279 3.9790 7.5999 15.7851 23.6740 Market leverage 0.4 0.0243 0.9330 3.7700 7.8166 12.2781 21.0071 28.7355 0.5 0.2987 2.8144 7.1155 11.9300 16.6855 25.3360 32.6755 0.6 1.3750 5.6615 10.8029 15.8607 20.5773 28.8424 35.7118 0.7 3.5755 9.0054 14.3865 19.3400 23.8277 31.5562 37.9315 B. Extended Merton model |$\sigma_v $| 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.1 0.0000 0.0000 0.0000 0.0000 0.0002 0.0695 1.0461 0.0000 0.0000 0.0001 0.0174 0.2339 2.7979 8.4882 0.2 0.0000 0.0000 0.0000 0.0001 0.0099 0.5905 3.5871 0.0000 0.0004 0.0917 0.9799 3.3836 11.3312 20.1151 0.3 0.0000 0.0000 0.0000 0.0028 0.0696 1.5494 6.1945 0.0000 0.0679 1.3394 4.9732 10.0149 20.4587 29.3097 Market leverage 0.4 0.0000 0.0000 0.0001 0.0159 0.2121 2.6408 8.3368 0.0047 0.9530 5.3511 11.6261 17.8715 28.2927 36.0544 0.5 0.0000 0.0000 0.0008 0.0467 0.4153 3.6005 9.8621 0.1952 4.3125 11.9132 19.2059 25.2896 34.3312 40.6166 0.6 0.0000 0.0000 0.0026 0.0888 0.6138 4.2703 10.7585 1.8473 10.6608 19.5015 26.2566 31.3300 38.3678 43.1563 0.7 0.0000 0.0000 0.0048 0.1226 0.7392 4.5833 11.0669 7.0904 18.6201 26.5155 31.6959 35.2889 40.1597 43.7222 Panel A reports hedge ratios for the Merton model, |$\frac{\partial \ln B}{\partial \ln E}$|⁠. The Merton hedge ratio follows from Equation (3). Panel B reports extended Merton model hedge ratios. Parameters are from Table 1. Hedge ratios are reported in percentage. Open in new tab The primary intuition for the numerical results from our extension of the Merton model is that it is exactly when firms have to make large repayments of debt where their financial situations can become particularly precarious. If a bond is de facto junior, debtholders know that if firm value is already low, the need to pay off a large amount of a de facto senior bond may push the firm close to bankruptcy prior to the payment of the de facto junior bond. That is, the firm may be continued when earlier debt is paid off, but this may put the firm in more jeopardy of being unable to pay off later debt if there are further negative shocks to the firm.15 If a bond is de facto senior, such a scenario is less relevant.16 Our choice to use an extended Merton model to illustrate the effects of de facto seniority stems from a goal of using as parsimonious a model as possible to illustrate the intuition. Nevertheless, we recognize that other choices can be made in the mechanisms of structural models of default. Thus, in Appendices B.1, B.2, and B.3, we consider additional structural models of default in an effort to better understand the effect of different assumptions on the de facto seniority effect. Although full details of our implementations are described in the respective appendices, we briefly explain the economic intuition behind each of the models here. In Appendix B.1, we consider the Geske (1977) model, where retiring debt is paid off by equityholders rather than out of firm value. Equityholders need to decide when each debt issue matures whether they want to pay-in to continue the firm. The effect of this discrete pay-in is to make equityholders consider the firm’s whole maturity structure when making a decision about continuing the firm. Even if only a small amount of debt is due today, equityholders may choose to default because they recognize that a large amount of debt is due in the future before they receive a payoff as residual claimants. Effectively, equityholders become less myopic about the firm. This has the effect of undoing the de facto seniority effect, predicting that de facto senior bonds could even have higher yield spreads and hedge ratios. This contrasts the extended Merton model, where equityholders do not have a pay-in and always want to continue the firm (if possible). In Appendix B.2, we extend the Geske model to allow for equityholders to pay-in an exogenously specified fraction of the face value of maturing debt. The Geske and extended Merton can be viewed as special cases of this model where the equity pay-ins are 100% and 0% of maturing debt, respectively. As would be expected, the de facto seniority effect weakens as the amount of equity pay-in increases. Finally, we consider a Leland and Toft (1996) model in Appendix B.3, a model in which there is endogenous equity pay-in, but only if there is a cash shortfall. In this model, firms continuously roll-over debt and equity pay-ins are only required if the proceeds from newly issued debt and from a fractional liquidation of firm assets are insufficient to cover coupons from all of the firm’s debt and the face value of maturing debt. This makes equity pay-ins infrequent and small. Though the Leland-Toft model makes the intuition significantly more complicated, the general effect is similar to an extended version of the Geske model with small equity pay-ins. There continues to be a de facto seniority effect, though one that is not quite as large as that in the extended Merton model. 2. Data and Summary Statistics The corporate bond data set used in this paper is the Financial Industry Regulatory Authority’s (FINRA) Trade Reporting and Compliance Engine (TRACE). This data set is the result of a regulatory initiative to increase price transparency in the corporate bond market and was implemented in three phases. On July 1, 2002, the first phase was implemented, requiring transaction information to be disseminated for investment-grade bonds with an issue size of $\$$ 1 billion or greater and 50 legacy high-yield bonds from the Fixed Income Pricing System (FIPS). Approximately 520 bonds were included in phase I. Phase II, which was fully implemented on April 14, 2003, increased the coverage to approximately 4,650 bonds and included smaller investment-grade issues. Phase III, which was fully implemented on February 7, 2005, covered approximately 99% of secondary bond market transactions. In our study, we use TRACE data on corporate bonds from July 2002 to December 2013. Bond transaction data from TRACE are merged with bond characteristics data from Mergent FISD, which allows us to determine bond characteristics such as age, maturity, amount outstanding, and rating. These data are then merged with equity return data from CRSP and accounting information from Compustat. Treasury data are obtained from the Constant Maturity Treasury (CMT) series. Mergent FISD is used to eliminate bonds that are putable, convertible, or have a fixed-price call option.17 CRSP is used to eliminate bonds issued by financial firms.18 In addition, we only keep institutional-sized trades, defined as trades of $\$$ 100,000 or greater in volume.19 To calculate yield spreads, we use the last trade for a bond in a month and calculate yields in a standard manner before subtracting off the yield of a similar maturity Treasury bond. TRACE does provide yields, but we find that yields from TRACE can occasionally be unreliable. The Treasury yield is based on the U.S. Department of Treasury’s Constant Maturity Treasury (CMT) series from which we linearly interpolate between provided maturities as necessary. As reported in Table 3, the mean yield spread in our sample is 1.81%, with means of 1.50% and 3.83% for investment-grade and speculative-grade bonds, respectively. As expected, yield spreads have an approximately 50% correlation with numerically coded ratings in the sample. Table 3 Bond summary statistics A. Yield spread sample . . All bonds . Investment grade . Speculative grade . Bonds 9,382 8,587 2,205 Bond-months 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Yield spread 1.81 1.33 1.57 1.50 1.18 1.21 3.83 3.45 2.08 Time-to-maturity 9.37 5.84 9.97 9.57 5.93 10.08 8.07 5.41 9.11 Age 4.86 3.83 4.00 4.73 3.70 3.95 5.74 4.82 4.20 Prop due prior 0.41 0.39 0.29 0.41 0.39 0.29 0.38 0.35 0.30 Amount out 554.11 400.00 553.08 578.38 400.00 572.68 395.20 300.00 363.68 Coupon 6.01 6.10 1.60 5.84 5.95 1.57 7.10 7.13 1.34 Rating 7.95 8.00 2.83 7.22 8.00 2.20 12.73 12.00 1.59 Amihud 1.17 0.53 1.78 1.12 0.51 1.72 1.47 0.63 2.11 Amihud vol 1.68 1.10 1.80 1.62 1.07 1.74 2.05 1.39 2.08 IRC 0.28 0.19 0.28 0.27 0.18 0.27 0.36 0.27 0.30 IRC vol 0.28 0.20 0.26 0.26 0.19 0.25 0.34 0.25 0.29 B. Hedge ratio sample All bonds Investment grade Speculative grade Bonds 7,373 6,611 1,614 Bond-months 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Bond return 0.50 0.38 2.70 0.47 0.35 2.52 0.69 0.67 3.67 Time-to-maturity 9.34 5.96 9.57 9.51 5.97 9.69 8.22 5.75 8.64 Age 4.46 3.49 3.75 4.32 3.35 3.68 5.40 4.51 4.04 Prop due prior 0.42 0.40 0.29 0.42 0.40 0.28 0.40 0.37 0.29 Amount out 628.00 500.00 576.11 655.40 500.00 596.35 447.06 350.00 370.82 Coupon 5.97 6.00 1.61 5.79 5.90 1.57 7.19 7.25 1.28 Rating 7.95 8.00 2.82 7.22 8.00 2.19 12.75 12.00 1.59 A. Yield spread sample . . All bonds . Investment grade . Speculative grade . Bonds 9,382 8,587 2,205 Bond-months 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Yield spread 1.81 1.33 1.57 1.50 1.18 1.21 3.83 3.45 2.08 Time-to-maturity 9.37 5.84 9.97 9.57 5.93 10.08 8.07 5.41 9.11 Age 4.86 3.83 4.00 4.73 3.70 3.95 5.74 4.82 4.20 Prop due prior 0.41 0.39 0.29 0.41 0.39 0.29 0.38 0.35 0.30 Amount out 554.11 400.00 553.08 578.38 400.00 572.68 395.20 300.00 363.68 Coupon 6.01 6.10 1.60 5.84 5.95 1.57 7.10 7.13 1.34 Rating 7.95 8.00 2.83 7.22 8.00 2.20 12.73 12.00 1.59 Amihud 1.17 0.53 1.78 1.12 0.51 1.72 1.47 0.63 2.11 Amihud vol 1.68 1.10 1.80 1.62 1.07 1.74 2.05 1.39 2.08 IRC 0.28 0.19 0.28 0.27 0.18 0.27 0.36 0.27 0.30 IRC vol 0.28 0.20 0.26 0.26 0.19 0.25 0.34 0.25 0.29 B. Hedge ratio sample All bonds Investment grade Speculative grade Bonds 7,373 6,611 1,614 Bond-months 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Bond return 0.50 0.38 2.70 0.47 0.35 2.52 0.69 0.67 3.67 Time-to-maturity 9.34 5.96 9.57 9.51 5.97 9.69 8.22 5.75 8.64 Age 4.46 3.49 3.75 4.32 3.35 3.68 5.40 4.51 4.04 Prop due prior 0.42 0.40 0.29 0.42 0.40 0.28 0.40 0.37 0.29 Amount out 628.00 500.00 576.11 655.40 500.00 596.35 447.06 350.00 370.82 Coupon 5.97 6.00 1.61 5.79 5.90 1.57 7.19 7.25 1.28 Rating 7.95 8.00 2.82 7.22 8.00 2.19 12.75 12.00 1.59 This table reports bond characteristics for our sample. Panel A reports the bonds in our yield spread analysis and panel B the bonds in our hedge ratio analysis. The mean, median, and standard deviations are reported for the whole sample, the investment grade subsample, and the speculative grade subsample. The number of bonds in the investment grade plus speculative grade subsamples is greater than the total number of bonds because some bonds are investment grade for part of the sample and speculative grade for part of the sample. Yield spread is the difference between the yield of a bond and a Treasury with similar maturity from the Constant Maturity Treasury (CMT) series in percentage. Time-to-maturity is reported in years. Age is the number of years since issuance. Prop due prior is the proportion of an issuer’s face value of debt due prior to a particular bond. Amount out is the face value outstanding in $\$$mm. Coupon is the coupon rate in percentage. Rating is a bond’s rating where Aaa=1 and C=21, with intermediate ratings also coded. Amihud, Amihud vol, IRC, and IRC vol are the Amihud measure, the volatility of the Amihud measure, implied round-trip cost, and the volatility of implied round-trip cost, respectively. All four are calculated following Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100. Bond return is the monthly log return of a bond in percentage. Open in new tab Table 3 Bond summary statistics A. Yield spread sample . . All bonds . Investment grade . Speculative grade . Bonds 9,382 8,587 2,205 Bond-months 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Yield spread 1.81 1.33 1.57 1.50 1.18 1.21 3.83 3.45 2.08 Time-to-maturity 9.37 5.84 9.97 9.57 5.93 10.08 8.07 5.41 9.11 Age 4.86 3.83 4.00 4.73 3.70 3.95 5.74 4.82 4.20 Prop due prior 0.41 0.39 0.29 0.41 0.39 0.29 0.38 0.35 0.30 Amount out 554.11 400.00 553.08 578.38 400.00 572.68 395.20 300.00 363.68 Coupon 6.01 6.10 1.60 5.84 5.95 1.57 7.10 7.13 1.34 Rating 7.95 8.00 2.83 7.22 8.00 2.20 12.73 12.00 1.59 Amihud 1.17 0.53 1.78 1.12 0.51 1.72 1.47 0.63 2.11 Amihud vol 1.68 1.10 1.80 1.62 1.07 1.74 2.05 1.39 2.08 IRC 0.28 0.19 0.28 0.27 0.18 0.27 0.36 0.27 0.30 IRC vol 0.28 0.20 0.26 0.26 0.19 0.25 0.34 0.25 0.29 B. Hedge ratio sample All bonds Investment grade Speculative grade Bonds 7,373 6,611 1,614 Bond-months 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Bond return 0.50 0.38 2.70 0.47 0.35 2.52 0.69 0.67 3.67 Time-to-maturity 9.34 5.96 9.57 9.51 5.97 9.69 8.22 5.75 8.64 Age 4.46 3.49 3.75 4.32 3.35 3.68 5.40 4.51 4.04 Prop due prior 0.42 0.40 0.29 0.42 0.40 0.28 0.40 0.37 0.29 Amount out 628.00 500.00 576.11 655.40 500.00 596.35 447.06 350.00 370.82 Coupon 5.97 6.00 1.61 5.79 5.90 1.57 7.19 7.25 1.28 Rating 7.95 8.00 2.82 7.22 8.00 2.19 12.75 12.00 1.59 A. Yield spread sample . . All bonds . Investment grade . Speculative grade . Bonds 9,382 8,587 2,205 Bond-months 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Yield spread 1.81 1.33 1.57 1.50 1.18 1.21 3.83 3.45 2.08 Time-to-maturity 9.37 5.84 9.97 9.57 5.93 10.08 8.07 5.41 9.11 Age 4.86 3.83 4.00 4.73 3.70 3.95 5.74 4.82 4.20 Prop due prior 0.41 0.39 0.29 0.41 0.39 0.29 0.38 0.35 0.30 Amount out 554.11 400.00 553.08 578.38 400.00 572.68 395.20 300.00 363.68 Coupon 6.01 6.10 1.60 5.84 5.95 1.57 7.10 7.13 1.34 Rating 7.95 8.00 2.83 7.22 8.00 2.20 12.73 12.00 1.59 Amihud 1.17 0.53 1.78 1.12 0.51 1.72 1.47 0.63 2.11 Amihud vol 1.68 1.10 1.80 1.62 1.07 1.74 2.05 1.39 2.08 IRC 0.28 0.19 0.28 0.27 0.18 0.27 0.36 0.27 0.30 IRC vol 0.28 0.20 0.26 0.26 0.19 0.25 0.34 0.25 0.29 B. Hedge ratio sample All bonds Investment grade Speculative grade Bonds 7,373 6,611 1,614 Bond-months 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Bond return 0.50 0.38 2.70 0.47 0.35 2.52 0.69 0.67 3.67 Time-to-maturity 9.34 5.96 9.57 9.51 5.97 9.69 8.22 5.75 8.64 Age 4.46 3.49 3.75 4.32 3.35 3.68 5.40 4.51 4.04 Prop due prior 0.42 0.40 0.29 0.42 0.40 0.28 0.40 0.37 0.29 Amount out 628.00 500.00 576.11 655.40 500.00 596.35 447.06 350.00 370.82 Coupon 5.97 6.00 1.61 5.79 5.90 1.57 7.19 7.25 1.28 Rating 7.95 8.00 2.82 7.22 8.00 2.19 12.75 12.00 1.59 This table reports bond characteristics for our sample. Panel A reports the bonds in our yield spread analysis and panel B the bonds in our hedge ratio analysis. The mean, median, and standard deviations are reported for the whole sample, the investment grade subsample, and the speculative grade subsample. The number of bonds in the investment grade plus speculative grade subsamples is greater than the total number of bonds because some bonds are investment grade for part of the sample and speculative grade for part of the sample. Yield spread is the difference between the yield of a bond and a Treasury with similar maturity from the Constant Maturity Treasury (CMT) series in percentage. Time-to-maturity is reported in years. Age is the number of years since issuance. Prop due prior is the proportion of an issuer’s face value of debt due prior to a particular bond. Amount out is the face value outstanding in $\$$mm. Coupon is the coupon rate in percentage. Rating is a bond’s rating where Aaa=1 and C=21, with intermediate ratings also coded. Amihud, Amihud vol, IRC, and IRC vol are the Amihud measure, the volatility of the Amihud measure, implied round-trip cost, and the volatility of implied round-trip cost, respectively. All four are calculated following Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100. Bond return is the monthly log return of a bond in percentage. Open in new tab Equity returns can be directly obtained from CRSP, but both corporate bond and Treasury returns need to be calculated. To calculate monthly corporate bond returns, we use the last trade for a bond in a month and calculate log returns as \begin{align} r_{t+1} = \ln\left(\frac{P_{t+1} + AI_{t+1} + C_{t+1}}{P_t + AI_t}\right), \end{align}(6) where |$P_{t+1}$| is a clean price, |$AI_{t+1}$| is the accrued interest, and |$C_{t+1}$| is the coupon paid during the month (if applicable). Unlike equities, many corporate bonds do not trade every day. Thus, the returns that we calculate for a bond in a particular month may not necessarily be based exactly on month-end to month-end prices. To account for this, we make two adjustments. First, we match Treasury and equity returns to the exact dates that are used to calculate bond returns. This eliminates any issues of nonsynchronization as our goal is to compare contemporaneous bond, equity, and Treasury returns in our hedge ratio analysis. Second, we scale all log returns to make them monthly. For example, if a log bond return is calculated over 25 business days, we scale the return by |$\frac{21}{25}$| to make it comparable to a return that actually occurs over exactly a one month period (⁠|$\approx$| 21 business days).20 The mean monthly corporate bond return in our sample is 0.50% per month, with means of 0.47% and 0.69% for investment-grade and speculative-grade bonds, respectively. As calculating bond returns requires bond prices in consecutive months, our bond return sample, which will be used later to calculate hedge ratios, is more restrictive than our yield spread sample. To calculate Treasury returns, we again use yields from the CMT series. The provided yields are par yields for on-the-run (and hence, liquid) bonds. Because the yields are par yields, we can calculate the return for a bond that trades at par at the start of a month by discounting cash flows using yields at the end of the month. Only discrete maturities are provided in the CMT series. Thus, we interpolate between maturities to calculate the return for a Treasury with the same maturity as the corporate bond being priced. Table 3 also reports other summary statistics for our corporate bond sample. Our yield spread sample covers 9,382 distinct bonds and 301,291 bond-months, whereas our hedge ratio sample is somewhat smaller. The vast majority of our sample covers investment-grade bonds, with over 85% of our bond-months coming from the investment-grade sample. As bond-level characteristics do not differ much between the yield spread and hedge ratio samples, we will largely describe the yield spread sample. The average time-to-maturity in our sample is 9.37 years, with a slightly higher mean for investment-grade (9.57 years) than speculative-grade (8.07 years) bonds. The median is lower at 5.84 years, reflecting the fact that time-to-maturity is positively skewed. The average age of bonds in our sample is close to 5 years. To measure how de facto senior a bond is in its firm’s maturity structure, we calculate the proportion of the issuer’s total face value of current outstanding debt from FISD that has a shorter remaining maturity than the bond being studied. Note that this includes all debt in Mergent FISD, not just bonds traded in TRACE. We call this variable proportion due prior and find that its average value is 0.41 in our sample. For amount outstanding and coupons, we observe significant differences between investment-grade and speculative-grade bonds. Investment-grade bond issues have an average amount outstanding of $\$$ 578mm, whereas speculative-grade bond issues have an average amount outstanding of $\$$395mm, about two-thirds the size of investment-grade issues. The smaller issue sizes for speculative-grade bonds reflects the fact that poorer rated firms tend to be smaller. Average coupons are also higher for speculative-grade bonds at 7.10% as compared to 5.84% for investment-grade bonds, reflecting the fact that most bonds are issued at par. We also report summary statistics for four bond-level illiquidity measures (the Amihud measure, Amihud volatility, the IRC measure, and IRC volatility) in panel A. All four variables are scaled by 100 and show similar distributional statistics as Dick-Nielsen, Feldhutter, and Lando (2012). Overall, speculative-grade bonds are less liquid than investment-grade bonds. In Table 4, we report summary stats for firm-level variables. We have fewer than 800 unique firms in our sample, reflecting the fact that many firms, even those with publicly traded equity, do not issue corporate bonds.21 The mean equity market capitalization of firms in our sample is $\$$ 46bn, reflecting the large size of corporate bond issuers. There is a drastic difference in the average size of investment-grade and speculative-grade issuers as investment-grade issuers average $\$$52bn in equity market capitalization compared to $\$$9bn for speculative-grade issuers. Investment-grade issuers also exhibit lower average equity volatility (28.00%) and leverage (30.25%) than speculative-grade issuers (40.44% equity volatility and 50.16% leverage). The higher numbers for speculative-grade firms are natural, as both variables reflect the higher credit risk of speculative-grade firms. Leverage is a direct measure of the amount of debt that a firm has issued, relative to firm value. Equity volatility reflects a combination of the leverage and the underlying asset volatility of the firm, both of which are positively related to default probabilities. In our empirical analysis, we include equity volatility as a credit risk control variable. In our sample, equity volatility has a significant 30% correlation with numerically coded ratings, suggesting that although there is significant overlap in credit risk as measured by the two variables, additional information can be provided by equity volatilities. Table 4 Firm summary statistics A. Yield spread sample . . All firms . Investment grade . Speculative grade . Firms 757 643 290 Firm-months 53,900 44,087 11,015 Observations 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Equity market cap 46.44 21.09 64.77 52.20 25.72 67.62 8.65 5.90 8.91 Equity vol 29.64 25.76 14.82 28.00 24.50 13.54 40.44 36.35 17.97 Pseudo-leverage 32.89 29.48 19.87 30.25 26.90 18.17 50.16 49.13 21.72 BE/ME 0.51 0.44 0.34 0.48 0.42 0.30 0.75 0.62 0.51 Firm maturity 10.58 10.37 4.16 10.79 10.62 4.19 9.18 9.01 3.66 B. Hedge ratio sample All firms Investment grade Speculative grade Firms 734 617 273 Firm-months 48,483 39,662 9,785 Observations 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Equity return 0.78 1.26 9.03 0.86 1.29 8.10 0.22 0.96 13.65 Equity market cap 47.04 22.51 62.92 52.84 27.12 65.49 8.66 5.81 9.18 Equity vol 29.90 25.92 15.03 28.27 24.73 13.70 40.64 36.16 18.57 Pseudo-leverage 32.03 28.91 18.92 29.40 26.34 17.27 49.39 49.02 20.17 BE/ME 0.50 0.44 0.33 0.47 0.42 0.29 0.72 0.61 0.47 Firm maturity 10.58 10.38 4.14 10.80 10.65 4.15 9.13 8.80 3.72 A. Yield spread sample . . All firms . Investment grade . Speculative grade . Firms 757 643 290 Firm-months 53,900 44,087 11,015 Observations 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Equity market cap 46.44 21.09 64.77 52.20 25.72 67.62 8.65 5.90 8.91 Equity vol 29.64 25.76 14.82 28.00 24.50 13.54 40.44 36.35 17.97 Pseudo-leverage 32.89 29.48 19.87 30.25 26.90 18.17 50.16 49.13 21.72 BE/ME 0.51 0.44 0.34 0.48 0.42 0.30 0.75 0.62 0.51 Firm maturity 10.58 10.37 4.16 10.79 10.62 4.19 9.18 9.01 3.66 B. Hedge ratio sample All firms Investment grade Speculative grade Firms 734 617 273 Firm-months 48,483 39,662 9,785 Observations 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Equity return 0.78 1.26 9.03 0.86 1.29 8.10 0.22 0.96 13.65 Equity market cap 47.04 22.51 62.92 52.84 27.12 65.49 8.66 5.81 9.18 Equity vol 29.90 25.92 15.03 28.27 24.73 13.70 40.64 36.16 18.57 Pseudo-leverage 32.03 28.91 18.92 29.40 26.34 17.27 49.39 49.02 20.17 BE/ME 0.50 0.44 0.33 0.47 0.42 0.29 0.72 0.61 0.47 Firm maturity 10.58 10.38 4.14 10.80 10.65 4.15 9.13 8.80 3.72 This table reports firm characteristics for our sample. Panel A reports firms in our yield spread analysis and panel B the firms in our hedge ratio analysis. Equity market cap is the equity market capitalization in $\$$bn. Equity vol is the volatility of a firm’s equity using daily log returns over the past year, annualized in percentage. Pseudo-leverage is the sum of debt in current liabilities (DLC) and Long-Term Debt (DLTT) divided by DLC + DLTT + equity market cap. BE/ME is the book value of equity divided by the market value of equity like in Asness and Frazzini (2013). Equity return is the monthly log equity return in percentage. Note that the sum of investment grade and speculative grade firm-months is slightly higher than total firm months because, in rare cases, a firm may have both an investment grade and speculative grade bond. Open in new tab Table 4 Firm summary statistics A. Yield spread sample . . All firms . Investment grade . Speculative grade . Firms 757 643 290 Firm-months 53,900 44,087 11,015 Observations 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Equity market cap 46.44 21.09 64.77 52.20 25.72 67.62 8.65 5.90 8.91 Equity vol 29.64 25.76 14.82 28.00 24.50 13.54 40.44 36.35 17.97 Pseudo-leverage 32.89 29.48 19.87 30.25 26.90 18.17 50.16 49.13 21.72 BE/ME 0.51 0.44 0.34 0.48 0.42 0.30 0.75 0.62 0.51 Firm maturity 10.58 10.37 4.16 10.79 10.62 4.19 9.18 9.01 3.66 B. Hedge ratio sample All firms Investment grade Speculative grade Firms 734 617 273 Firm-months 48,483 39,662 9,785 Observations 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Equity return 0.78 1.26 9.03 0.86 1.29 8.10 0.22 0.96 13.65 Equity market cap 47.04 22.51 62.92 52.84 27.12 65.49 8.66 5.81 9.18 Equity vol 29.90 25.92 15.03 28.27 24.73 13.70 40.64 36.16 18.57 Pseudo-leverage 32.03 28.91 18.92 29.40 26.34 17.27 49.39 49.02 20.17 BE/ME 0.50 0.44 0.33 0.47 0.42 0.29 0.72 0.61 0.47 Firm maturity 10.58 10.38 4.14 10.80 10.65 4.15 9.13 8.80 3.72 A. Yield spread sample . . All firms . Investment grade . Speculative grade . Firms 757 643 290 Firm-months 53,900 44,087 11,015 Observations 301,291 261,372 39,919 Mean Med SD Mean Med SD Mean Med SD Equity market cap 46.44 21.09 64.77 52.20 25.72 67.62 8.65 5.90 8.91 Equity vol 29.64 25.76 14.82 28.00 24.50 13.54 40.44 36.35 17.97 Pseudo-leverage 32.89 29.48 19.87 30.25 26.90 18.17 50.16 49.13 21.72 BE/ME 0.51 0.44 0.34 0.48 0.42 0.30 0.75 0.62 0.51 Firm maturity 10.58 10.37 4.16 10.79 10.62 4.19 9.18 9.01 3.66 B. Hedge ratio sample All firms Investment grade Speculative grade Firms 734 617 273 Firm-months 48,483 39,662 9,785 Observations 239,862 208,317 31,545 Mean Med SD Mean Med SD Mean Med SD Equity return 0.78 1.26 9.03 0.86 1.29 8.10 0.22 0.96 13.65 Equity market cap 47.04 22.51 62.92 52.84 27.12 65.49 8.66 5.81 9.18 Equity vol 29.90 25.92 15.03 28.27 24.73 13.70 40.64 36.16 18.57 Pseudo-leverage 32.03 28.91 18.92 29.40 26.34 17.27 49.39 49.02 20.17 BE/ME 0.50 0.44 0.33 0.47 0.42 0.29 0.72 0.61 0.47 Firm maturity 10.58 10.38 4.14 10.80 10.65 4.15 9.13 8.80 3.72 This table reports firm characteristics for our sample. Panel A reports firms in our yield spread analysis and panel B the firms in our hedge ratio analysis. Equity market cap is the equity market capitalization in $\$$bn. Equity vol is the volatility of a firm’s equity using daily log returns over the past year, annualized in percentage. Pseudo-leverage is the sum of debt in current liabilities (DLC) and Long-Term Debt (DLTT) divided by DLC + DLTT + equity market cap. BE/ME is the book value of equity divided by the market value of equity like in Asness and Frazzini (2013). Equity return is the monthly log equity return in percentage. Note that the sum of investment grade and speculative grade firm-months is slightly higher than total firm months because, in rare cases, a firm may have both an investment grade and speculative grade bond. Open in new tab In addition to equity volatility, we also include book equity-to-market equity (BE/ME) as a measure of firm-level credit quality. The empirical literature on equity returns has found BE/ME to be related to the relative strength of a firm’s economic fundamentals (Lakonishok, Shleifer, and Vishny 1994; Fama and French 1995). Furthermore, Fama and French (1992) show that BE/ME captures the effect of leverage, an important structural model input. Though there is much debate as to whether the return premium associated with book-to-market ratio is due to risk or mispricing (see, e.g., Fama and French 1993, 1996; Lakonishok, Shleifer, and Vishny 1994; Haugen 1995; MacKinlay 1995; and Daniel and Titman 1997), our use of the book-to-market ratio is much more basic. We simply take it as reflecting the market’s assessment of a firm’s strength. We construct BE/ME in a similar manner to Asness and Frazzini (2013), who argue that it is important to use more timely stock price data relative to book data in order to better forecast future book-to-market ratio. For the book value of equity, we use the book value of common equity (CEQ) from Compustat and assume that it is publicly available three months after the fiscal year end. For the market value of equity, we use common shares outstanding (CSHO) times price at the end of the fiscal year (PRCC_F), but adjust this using returns without dividends from CRSP. For example, to calculate BE/ME for a firm in May 2010 whose last fiscal year ended in December 2009, we take the book equity and market equity (CSHO |$\times$| PRCC_F) from Compustat at of the end of the 2009 fiscal year and scale the market equity by (1 + r), where r is the return of the firm’s equity without dividends from December 2009 to May 2010.22 We find that BE/ME is almost 60% higher for speculative-grade bonds as than for investment-grade bonds. We also note that BE/ME is significantly correlated with both numerically coded ratings and equity volatility, with correlations around 30%–40%. The final firm-level variable that we report is the average firm-level maturity. To construct this variable, we take the amount outstanding-weighted average of the time-to-maturity of an issuer’s bonds. This variable is used to capture the fact that some firms tend to issue bonds at a shorter maturity and some at a longer maturity. The mean and median of firm-level average maturity is somewhat higher than the reported average bond maturity in Table 3, because there is a positive correlation between the time-to-maturity and the amount outstanding. 3. Empirical Results 3.1 Yield spreads We first discuss a case in our data to highlight the de facto seniority effect before turning to formal empirical analysis using regressions. In 2013, Amgen had two bonds with very similar time-to-maturity, a $\$$ 1.25bn bond maturing on May 15, 2017 (“May bond”), and a $\$$1.1bn bond maturing on June 1, 2017 (“June bond”). During 2013, the two bonds also had similar levels of liquidity as measured by the Amihud (2002) measure and the implied round-trip cost measure of Feldhutter (2012). Nevertheless, during 2013, the June bond consistently traded at a yield spread higher than the May bond, with an average difference of 8 basis points. This was roughly 10% of the average yield spreads of the bonds during the year, despite the bonds having the same issuer, same rating, similar liquidity, and very close maturity. The primary reason for the yield spread difference between the bonds is that the sequential payment of debt means that, in the future, Amgen would have to pay off the May bond prior to paying off the June bond, hence making the May bond de facto senior to the June bond. To formally examine the relation between yield spreads and de facto seniority, we next turn to a standard regression framework to examine a broader sample, while controlling for differences across bonds. Our empirical specification for examining the effect of de facto seniority on yield spreads is \begin{align} \text{yield spread}_{it} = \alpha_t + \beta \times \text{proportion due prior}_{it} + \delta X_{it} + \varepsilon_{it}, \end{align}(7) where |$X_{it}$| is a set of control variables that includes credit controls, illiquidity controls, firm- and bond-level maturity, and firm fixed effects. We also include time fixed effects to account for differences in yield spreads due to changes in macroeconomic conditions across time in our sample. Proportion due prior is our variable of interest. A lower proportion due prior corresponds to a bond that is earlier in its firm’s maturity structure and therefore is de facto senior. In the extended Merton model examples in Section 1, we compare model-implied yield spreads for two hypothetical bonds with proportions due prior equal to 0.1 and 0.8. In our baseline specification, reported in Column (1) of Table 5, panel A, we control only for a bond’s maturity (T) and the bond-level Moody’s credit rating (Rating).23 We include the bond’s maturity to ensure that proportion due prior is not simply proxying for a pure maturity effect. The coefficient on proportion due prior is a statistically and economically significant 95 basis points. All standard errors are two-way clustered by firm and time, following Cameron, Gelbach, and Miller (2011). Like Dick-Nielsen, Feldhutter, and Lando (2012), this allows for time series correlation (for all bonds from the same issuer), cross-correlation, and heteroscedasticity. To gauge the economic magnitude of the 95-basis-point coefficient on proportion due prior, we note that a one-notch decline in credit rating is associated with a 30-basis-point increase in yield spreads. Thus, the de facto seniority effect is equivalent to a decline in credit rating of more than three notches. Table 5 Yield spreads A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . . Full . Full . Full . Full . Full . IG . Junk . No crisis . Proportion due prior 0.95 0.81 0.70 0.48 0.42 0.34 0.74 0.50 [5.49] [6.57] [7.07] [6.74] [5.84] [5.35] [5.60] [7.35] T –0.0054 –0.0004 0.0039 0.0037 0.0041 0.0078 –0.0013 0.0058 [–0.98] [–0.10] [1.58] [1.87] [1.90] [4.11] [–0.28] [2.81] Rating 0.30 0.21 0.21 0.22 0.22 0.15 0.29 0.21 [9.70] [8.40] [8.62] [10.87] [11.12] [9.29] [7.61] [11.01] BE/ME 1.02 1.04 0.87 0.81 0.70 0.97 0.72 [5.44] [5.50] [7.26] [8.04] [7.64] [6.74] [6.16] Equity vol 0.0300 0.0292 0.0262 0.0252 0.0184 0.0160 0.0220 [11.39] [11.45] [11.02] [10.19] [8.67] [2.50] [8.88] Firm T –0.01 –0.02 –0.02 –0.01 0.00 –0.01 [–2.33] [–3.06] [–3.23] [–2.30] [0.18] [–2.29] Amihud 0.06 0.05 0.05 0.08 0.08 [4.38] [2.77] [4.35] [2.88] [3.66] IRC 0.74 0.90 0.77 0.73 0.87 [13.65] [10.56] [9.79] [4.98] [9.64] Amihud vol 0.04 0.03 0.07 0.04 [6.99] [5.89] [5.59] [5.36] IRC vol –0.03 –0.06 0.21 0.01 [–0.48] [–1.50] [1.51] [0.19] |$R^2$| 0.62 0.71 0.71 0.74 0.75 0.72 0.73 0.68 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 B. With firm fixed effects (1) (2) (3) (4) (5) (6) (7) (8) Full Full Full Full Full IG Junk No crisis Proportion due prior 0.70 0.67 0.66 0.54 0.48 0.47 0.85 0.57 [12.33] [11.70] [11.49] [10.72] [9.46] [11.32] [6.53] [10.95] T 0.0045 0.0060 0.0062 0.0063 0.0071 0.0070 0.0059 0.0084 [3.89] [5.85] [6.46] [5.59] [5.70] [5.93] [1.76] [6.28] Rating 0.27 0.20 0.20 0.21 0.21 0.12 0.23 0.20 [9.48] [9.52] [9.56] [9.91] [9.89] [9.59] [5.39] [8.45] BE/ME 1.20 1.20 1.14 1.12 1.02 1.10 0.97 [11.00] [11.01] [12.49] [13.06] [9.72] [9.16] [9.17] Equity vol 0.0326 0.0326 0.0310 0.0302 0.0227 0.0214 0.0265 [10.14] [10.04] [10.21] [9.91] [7.47] [3.99] [8.84] Firm T –0.00 –0.01 –0.01 0.00 0.03 –0.00 [–0.27] [–0.95] [–1.21] [0.26] [1.73] [–0.61] Amihud 0.03 0.03 0.04 0.03 0.05 [5.12] [3.05] [5.07] [2.78] [4.26] IRC 0.40 0.51 0.46 0.36 0.48 [10.81] [8.91] [7.84] [3.95] [7.64] Amihud vol 0.02 0.02 0.05 0.02 [6.18] [4.99] [5.74] [5.00] IRC vol –0.04 –0.06 0.18 –0.01 [–0.93] [–1.82] [2.47] [–0.19] |$R^2$| 0.76 0.80 0.80 0.82 0.83 0.79 0.82 0.79 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . . Full . Full . Full . Full . Full . IG . Junk . No crisis . Proportion due prior 0.95 0.81 0.70 0.48 0.42 0.34 0.74 0.50 [5.49] [6.57] [7.07] [6.74] [5.84] [5.35] [5.60] [7.35] T –0.0054 –0.0004 0.0039 0.0037 0.0041 0.0078 –0.0013 0.0058 [–0.98] [–0.10] [1.58] [1.87] [1.90] [4.11] [–0.28] [2.81] Rating 0.30 0.21 0.21 0.22 0.22 0.15 0.29 0.21 [9.70] [8.40] [8.62] [10.87] [11.12] [9.29] [7.61] [11.01] BE/ME 1.02 1.04 0.87 0.81 0.70 0.97 0.72 [5.44] [5.50] [7.26] [8.04] [7.64] [6.74] [6.16] Equity vol 0.0300 0.0292 0.0262 0.0252 0.0184 0.0160 0.0220 [11.39] [11.45] [11.02] [10.19] [8.67] [2.50] [8.88] Firm T –0.01 –0.02 –0.02 –0.01 0.00 –0.01 [–2.33] [–3.06] [–3.23] [–2.30] [0.18] [–2.29] Amihud 0.06 0.05 0.05 0.08 0.08 [4.38] [2.77] [4.35] [2.88] [3.66] IRC 0.74 0.90 0.77 0.73 0.87 [13.65] [10.56] [9.79] [4.98] [9.64] Amihud vol 0.04 0.03 0.07 0.04 [6.99] [5.89] [5.59] [5.36] IRC vol –0.03 –0.06 0.21 0.01 [–0.48] [–1.50] [1.51] [0.19] |$R^2$| 0.62 0.71 0.71 0.74 0.75 0.72 0.73 0.68 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 B. With firm fixed effects (1) (2) (3) (4) (5) (6) (7) (8) Full Full Full Full Full IG Junk No crisis Proportion due prior 0.70 0.67 0.66 0.54 0.48 0.47 0.85 0.57 [12.33] [11.70] [11.49] [10.72] [9.46] [11.32] [6.53] [10.95] T 0.0045 0.0060 0.0062 0.0063 0.0071 0.0070 0.0059 0.0084 [3.89] [5.85] [6.46] [5.59] [5.70] [5.93] [1.76] [6.28] Rating 0.27 0.20 0.20 0.21 0.21 0.12 0.23 0.20 [9.48] [9.52] [9.56] [9.91] [9.89] [9.59] [5.39] [8.45] BE/ME 1.20 1.20 1.14 1.12 1.02 1.10 0.97 [11.00] [11.01] [12.49] [13.06] [9.72] [9.16] [9.17] Equity vol 0.0326 0.0326 0.0310 0.0302 0.0227 0.0214 0.0265 [10.14] [10.04] [10.21] [9.91] [7.47] [3.99] [8.84] Firm T –0.00 –0.01 –0.01 0.00 0.03 –0.00 [–0.27] [–0.95] [–1.21] [0.26] [1.73] [–0.61] Amihud 0.03 0.03 0.04 0.03 0.05 [5.12] [3.05] [5.07] [2.78] [4.26] IRC 0.40 0.51 0.46 0.36 0.48 [10.81] [8.91] [7.84] [3.95] [7.64] Amihud vol 0.02 0.02 0.05 0.02 [6.18] [4.99] [5.74] [5.00] IRC vol –0.04 –0.06 0.18 –0.01 [–0.93] [–1.82] [2.47] [–0.19] |$R^2$| 0.76 0.80 0.80 0.82 0.83 0.79 0.82 0.79 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 The dependent variable in this table is yield spread in percentage. Panel A reports results without firm fixed effects, and panel B includes firm fixed effects. All regressions include time fixed effects to control for time-varying macroeconomic conditions. Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. Rating is Moody’s rating coded as Aaa=1 and C=21, with intermediate ratings coded. BE/ME is book equity divided by market equity for the bond issuer, expressed in decimals. Equity vol is an annualized equity volatility calculated from daily returns in the last year in percentage. Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. Amihud, Amihud vol, IRC, and IRC vol are the Amihud measure, the volatility of the Amihud measure, implied round-trip cost, and the volatility of implied round-trip cost, respectively. All four are calculated following Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Table 5 Yield spreads A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . . Full . Full . Full . Full . Full . IG . Junk . No crisis . Proportion due prior 0.95 0.81 0.70 0.48 0.42 0.34 0.74 0.50 [5.49] [6.57] [7.07] [6.74] [5.84] [5.35] [5.60] [7.35] T –0.0054 –0.0004 0.0039 0.0037 0.0041 0.0078 –0.0013 0.0058 [–0.98] [–0.10] [1.58] [1.87] [1.90] [4.11] [–0.28] [2.81] Rating 0.30 0.21 0.21 0.22 0.22 0.15 0.29 0.21 [9.70] [8.40] [8.62] [10.87] [11.12] [9.29] [7.61] [11.01] BE/ME 1.02 1.04 0.87 0.81 0.70 0.97 0.72 [5.44] [5.50] [7.26] [8.04] [7.64] [6.74] [6.16] Equity vol 0.0300 0.0292 0.0262 0.0252 0.0184 0.0160 0.0220 [11.39] [11.45] [11.02] [10.19] [8.67] [2.50] [8.88] Firm T –0.01 –0.02 –0.02 –0.01 0.00 –0.01 [–2.33] [–3.06] [–3.23] [–2.30] [0.18] [–2.29] Amihud 0.06 0.05 0.05 0.08 0.08 [4.38] [2.77] [4.35] [2.88] [3.66] IRC 0.74 0.90 0.77 0.73 0.87 [13.65] [10.56] [9.79] [4.98] [9.64] Amihud vol 0.04 0.03 0.07 0.04 [6.99] [5.89] [5.59] [5.36] IRC vol –0.03 –0.06 0.21 0.01 [–0.48] [–1.50] [1.51] [0.19] |$R^2$| 0.62 0.71 0.71 0.74 0.75 0.72 0.73 0.68 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 B. With firm fixed effects (1) (2) (3) (4) (5) (6) (7) (8) Full Full Full Full Full IG Junk No crisis Proportion due prior 0.70 0.67 0.66 0.54 0.48 0.47 0.85 0.57 [12.33] [11.70] [11.49] [10.72] [9.46] [11.32] [6.53] [10.95] T 0.0045 0.0060 0.0062 0.0063 0.0071 0.0070 0.0059 0.0084 [3.89] [5.85] [6.46] [5.59] [5.70] [5.93] [1.76] [6.28] Rating 0.27 0.20 0.20 0.21 0.21 0.12 0.23 0.20 [9.48] [9.52] [9.56] [9.91] [9.89] [9.59] [5.39] [8.45] BE/ME 1.20 1.20 1.14 1.12 1.02 1.10 0.97 [11.00] [11.01] [12.49] [13.06] [9.72] [9.16] [9.17] Equity vol 0.0326 0.0326 0.0310 0.0302 0.0227 0.0214 0.0265 [10.14] [10.04] [10.21] [9.91] [7.47] [3.99] [8.84] Firm T –0.00 –0.01 –0.01 0.00 0.03 –0.00 [–0.27] [–0.95] [–1.21] [0.26] [1.73] [–0.61] Amihud 0.03 0.03 0.04 0.03 0.05 [5.12] [3.05] [5.07] [2.78] [4.26] IRC 0.40 0.51 0.46 0.36 0.48 [10.81] [8.91] [7.84] [3.95] [7.64] Amihud vol 0.02 0.02 0.05 0.02 [6.18] [4.99] [5.74] [5.00] IRC vol –0.04 –0.06 0.18 –0.01 [–0.93] [–1.82] [2.47] [–0.19] |$R^2$| 0.76 0.80 0.80 0.82 0.83 0.79 0.82 0.79 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . . Full . Full . Full . Full . Full . IG . Junk . No crisis . Proportion due prior 0.95 0.81 0.70 0.48 0.42 0.34 0.74 0.50 [5.49] [6.57] [7.07] [6.74] [5.84] [5.35] [5.60] [7.35] T –0.0054 –0.0004 0.0039 0.0037 0.0041 0.0078 –0.0013 0.0058 [–0.98] [–0.10] [1.58] [1.87] [1.90] [4.11] [–0.28] [2.81] Rating 0.30 0.21 0.21 0.22 0.22 0.15 0.29 0.21 [9.70] [8.40] [8.62] [10.87] [11.12] [9.29] [7.61] [11.01] BE/ME 1.02 1.04 0.87 0.81 0.70 0.97 0.72 [5.44] [5.50] [7.26] [8.04] [7.64] [6.74] [6.16] Equity vol 0.0300 0.0292 0.0262 0.0252 0.0184 0.0160 0.0220 [11.39] [11.45] [11.02] [10.19] [8.67] [2.50] [8.88] Firm T –0.01 –0.02 –0.02 –0.01 0.00 –0.01 [–2.33] [–3.06] [–3.23] [–2.30] [0.18] [–2.29] Amihud 0.06 0.05 0.05 0.08 0.08 [4.38] [2.77] [4.35] [2.88] [3.66] IRC 0.74 0.90 0.77 0.73 0.87 [13.65] [10.56] [9.79] [4.98] [9.64] Amihud vol 0.04 0.03 0.07 0.04 [6.99] [5.89] [5.59] [5.36] IRC vol –0.03 –0.06 0.21 0.01 [–0.48] [–1.50] [1.51] [0.19] |$R^2$| 0.62 0.71 0.71 0.74 0.75 0.72 0.73 0.68 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 B. With firm fixed effects (1) (2) (3) (4) (5) (6) (7) (8) Full Full Full Full Full IG Junk No crisis Proportion due prior 0.70 0.67 0.66 0.54 0.48 0.47 0.85 0.57 [12.33] [11.70] [11.49] [10.72] [9.46] [11.32] [6.53] [10.95] T 0.0045 0.0060 0.0062 0.0063 0.0071 0.0070 0.0059 0.0084 [3.89] [5.85] [6.46] [5.59] [5.70] [5.93] [1.76] [6.28] Rating 0.27 0.20 0.20 0.21 0.21 0.12 0.23 0.20 [9.48] [9.52] [9.56] [9.91] [9.89] [9.59] [5.39] [8.45] BE/ME 1.20 1.20 1.14 1.12 1.02 1.10 0.97 [11.00] [11.01] [12.49] [13.06] [9.72] [9.16] [9.17] Equity vol 0.0326 0.0326 0.0310 0.0302 0.0227 0.0214 0.0265 [10.14] [10.04] [10.21] [9.91] [7.47] [3.99] [8.84] Firm T –0.00 –0.01 –0.01 0.00 0.03 –0.00 [–0.27] [–0.95] [–1.21] [0.26] [1.73] [–0.61] Amihud 0.03 0.03 0.04 0.03 0.05 [5.12] [3.05] [5.07] [2.78] [4.26] IRC 0.40 0.51 0.46 0.36 0.48 [10.81] [8.91] [7.84] [3.95] [7.64] Amihud vol 0.02 0.02 0.05 0.02 [6.18] [4.99] [5.74] [5.00] IRC vol –0.04 –0.06 0.18 –0.01 [–0.93] [–1.82] [2.47] [–0.19] |$R^2$| 0.76 0.80 0.80 0.82 0.83 0.79 0.82 0.79 Observations 301,270 293,499 293,499 240,037 201,233 175,385 25,848 168,117 The dependent variable in this table is yield spread in percentage. Panel A reports results without firm fixed effects, and panel B includes firm fixed effects. All regressions include time fixed effects to control for time-varying macroeconomic conditions. Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. Rating is Moody’s rating coded as Aaa=1 and C=21, with intermediate ratings coded. BE/ME is book equity divided by market equity for the bond issuer, expressed in decimals. Equity vol is an annualized equity volatility calculated from daily returns in the last year in percentage. Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. Amihud, Amihud vol, IRC, and IRC vol are the Amihud measure, the volatility of the Amihud measure, implied round-trip cost, and the volatility of implied round-trip cost, respectively. All four are calculated following Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab In Columns (2) to (5) of Table 5, panel A, we add additional controls. In Column (2), we add book equity-to-market equity (BE/ME) and equity volatility, both of which are empirical proxies for a firm’s credit risk. In Column (3), we add the weighted-average maturity of all of a firm’s corporate bonds (firm T). This reflects firms’ endogenous choices to issue either debt with longer or shorter maturity. In Columns (4) and (5), we add the four main measures of illiquidity from Dick-Nielsen, Feldhutter, and Lando (2012): the Amihud measure, the implied round-trip cost (IRC) measure from Feldhutter (2012), and the volatilities of the two measures. Regardless of the specification, proportion due prior remains highly significant, with coefficients ranging from 42 to 81 basis points. We consider subsamples in Columns (6) to (8). In Columns (6) and (7), we rerun the regression from Column (5), but do so for the subset of investment-grade and speculative-grade bonds, respectively. For both samples, we continue to find a significant effect for proportion due prior, though the coefficient is larger for speculative-grade bonds at 74 basis points compared to 34 basis points for investment-grade bonds. Such a difference likely reflects the fact that being very late in the maturity structure of a very safe firm still means that a bond is relatively safe, whereas the difference between an early maturing and late maturing bond for a speculative-grade firm is particularly important. Such firms may become insolvent exactly after they have paid off the early maturing debt, but are unable to raise new funding through new debt issues. In Column (8), we exclude the crisis period (July 2007 – June 2009) and rerun our regression. We find that the effect of proportion due prior is 50 basis points when excluding the crisis, slightly higher than the 42 basis points (Column (5)) when including the crisis. Thus, the effect we find is not a financial crisis-specific effect. We include firm fixed effects in our specifications in panel B of Table 5. This controls for time-invariant firm-level effects including average credit quality and a firm’s propensity to issue shorter or longer-term debt. We continue to include credit ratings, book-to-market, and equity volatility as these variables allow for time variation in credit quality. Indeed, our results suggest that our time-varying credit controls remain important in measuring credit quality as evidenced by the similar coefficients on credit controls in panels A and B. We also continue to include the average firm-level maturity of debt as this potentially proxies for the time-varying component of a firm’s propensity to choose shorter- or longer-term debt. However, we find that this variable is much less significant when including firm fixed effects, suggesting that a firm’s propensity to issue longer- or shorter-term debt is not highly time varying. We also maintain illiquidity controls as illiquidity can be bond specific. Even with all of these controls, we continue to find a significant effect for proportion due prior. In particular, the coefficient on proportion due prior is 0.48 when including all controls (Column (5)), similar to the results without firm fixed effects. This effect is larger than the effect of decreasing credit rating by two notches (21 basis points per notch). We also continue to find subsample results similar to those in panel A. In Appendix C, we consider robustness checks where we use alternative credit controls such as asset volatility and model-implied leverage and also include their squared terms. Our results are similar. As an additional test of the de facto seniority effect, we consider paired sample analysis.24 We match bonds issued by the same firm with close maturities. For each bond, we match to bonds with maturities between one and 90 days later. Then, we calculate the difference in yield spreads and the difference in proportion due prior between the bonds. The de facto seniority effect implies that a greater difference in proportion due prior is associated with a greater difference in yield spreads, controlling for other differences between the bonds. Hence, our empirical specification is \begin{align}\label{eq:diffs} \text{Yield spread}_{jt} - \text{Yield spread}_{it} &= \alpha_t + \beta_1 (\text{prop due prior}_{jt} - \text{prop due prior}_{it})\nonumber\\ &\quad+ \beta_2 (T_{jt} - T_{it}) + \beta_3 (\text{Rating}_{jt} - \text{Rating}_{it})\nonumber\\ &\quad+ \beta_4 (\text{Liquidity}_{jt} - \text{Liquidity}_{it}) + \varepsilon_{jit}, \end{align}(8) where bond |$j$| is the later maturing bond in the pair. Restricting the pairs to bonds with very close maturity has two possible benefits. First, there is little variation in the maturity difference, making it unlikely that maturity differences can explain yield spread differences. Second, the correlation between proportion due prior differences and maturity differences is smaller than the correlation in levels. Both of these benefits could help with the identification of the proportion due prior effect, though with the caveat that we can only include a subsample of the bonds that were in the original sample. A further benefit of this matched sample analysis is that, because we match by issuer, firm-level effects drop out.25 In Table 6, we estimate specification (8), finding a de facto seniority effect of 81 basis points when controlling for differences in all four measures of illiquidity in addition to controlling for differences in time-to-maturity and rating. We note that this specification has only 25,576 observations as compared to 201,233 observations in the comparable regression in Table 5. When we further restrict our sample to cases where bonds |$i$| and |$j$| are close in illiquidity (Columns (3) and (4)),26 our sample drops further to fewer than 20,000 observation and we find a 104-basis-point de facto seniority effect. Thus, our results on de facto seniority are robust to an alternative matched sample analysis. Table 6 Matched sample analysis . (1) . (2) . (3) . (4) . Prop due prior|$_ {jt}$|- Prop due prior|$_{it}$| 0.86 0.81 1.06 1.04 [2.91] [3.01] [2.27] [2.38] T|$_{jt}$|- T|$_{it}$| –0.03 –0.21 0.03 –0.21 [–0.12] [–1.38] [0.13] [–1.07] Rating|$_{jt}$|- Rating|$_{it}$| 0.21 0.23 0.19 0.21 [3.11] [3.63] [2.62] [3.65] Amihud|$_{jt}$|- Amihud|$_{it}$| 0.03 0.04 0.01 0.01 [3.24] [3.08] [0.81] [1.06] IRC|$_{jt}$|- IRC|$_{it}$| 0.12 0.23 0.16 0.23 [5.11] [5.02] [5.88] [2.98] Amihud vol|$_{jt}$|- Amihud vol|$_{it}$| 0.02 0.01 [1.88] [1.64] IRC vol|$_{jt}$|- IRC vol|$_{it}$| –0.04 0.01 [–0.56] [0.14] |$R^2$| 0.06 0.11 0.02 0.04 Observations 44,283 25,576 19,365 11,969 . (1) . (2) . (3) . (4) . Prop due prior|$_ {jt}$|- Prop due prior|$_{it}$| 0.86 0.81 1.06 1.04 [2.91] [3.01] [2.27] [2.38] T|$_{jt}$|- T|$_{it}$| –0.03 –0.21 0.03 –0.21 [–0.12] [–1.38] [0.13] [–1.07] Rating|$_{jt}$|- Rating|$_{it}$| 0.21 0.23 0.19 0.21 [3.11] [3.63] [2.62] [3.65] Amihud|$_{jt}$|- Amihud|$_{it}$| 0.03 0.04 0.01 0.01 [3.24] [3.08] [0.81] [1.06] IRC|$_{jt}$|- IRC|$_{it}$| 0.12 0.23 0.16 0.23 [5.11] [5.02] [5.88] [2.98] Amihud vol|$_{jt}$|- Amihud vol|$_{it}$| 0.02 0.01 [1.88] [1.64] IRC vol|$_{jt}$|- IRC vol|$_{it}$| –0.04 0.01 [–0.56] [0.14] |$R^2$| 0.06 0.11 0.02 0.04 Observations 44,283 25,576 19,365 11,969 The sample in this table consists of matched pairs of bonds from the same issuer, with yield spreads measured in the same month. Paired bonds mature within 90 days of each other to minimize differential maturity effects. The dependent variable is the yield spread of the longer maturity bond minus the yield spread of the shorter maturity bond. The independent variables are similarly differences in variables, where Prop due prior, T, Rating, Amihud, IRC, Amihud vol, and IRC vol are defined in Table 5. In the first two columns, we match only on maturity (within firm-month). In Columns (3) and (4), we also restrict the matched bonds to have similar Amihud and IRC. Regressions include time fixed effects. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Table 6 Matched sample analysis . (1) . (2) . (3) . (4) . Prop due prior|$_ {jt}$|- Prop due prior|$_{it}$| 0.86 0.81 1.06 1.04 [2.91] [3.01] [2.27] [2.38] T|$_{jt}$|- T|$_{it}$| –0.03 –0.21 0.03 –0.21 [–0.12] [–1.38] [0.13] [–1.07] Rating|$_{jt}$|- Rating|$_{it}$| 0.21 0.23 0.19 0.21 [3.11] [3.63] [2.62] [3.65] Amihud|$_{jt}$|- Amihud|$_{it}$| 0.03 0.04 0.01 0.01 [3.24] [3.08] [0.81] [1.06] IRC|$_{jt}$|- IRC|$_{it}$| 0.12 0.23 0.16 0.23 [5.11] [5.02] [5.88] [2.98] Amihud vol|$_{jt}$|- Amihud vol|$_{it}$| 0.02 0.01 [1.88] [1.64] IRC vol|$_{jt}$|- IRC vol|$_{it}$| –0.04 0.01 [–0.56] [0.14] |$R^2$| 0.06 0.11 0.02 0.04 Observations 44,283 25,576 19,365 11,969 . (1) . (2) . (3) . (4) . Prop due prior|$_ {jt}$|- Prop due prior|$_{it}$| 0.86 0.81 1.06 1.04 [2.91] [3.01] [2.27] [2.38] T|$_{jt}$|- T|$_{it}$| –0.03 –0.21 0.03 –0.21 [–0.12] [–1.38] [0.13] [–1.07] Rating|$_{jt}$|- Rating|$_{it}$| 0.21 0.23 0.19 0.21 [3.11] [3.63] [2.62] [3.65] Amihud|$_{jt}$|- Amihud|$_{it}$| 0.03 0.04 0.01 0.01 [3.24] [3.08] [0.81] [1.06] IRC|$_{jt}$|- IRC|$_{it}$| 0.12 0.23 0.16 0.23 [5.11] [5.02] [5.88] [2.98] Amihud vol|$_{jt}$|- Amihud vol|$_{it}$| 0.02 0.01 [1.88] [1.64] IRC vol|$_{jt}$|- IRC vol|$_{it}$| –0.04 0.01 [–0.56] [0.14] |$R^2$| 0.06 0.11 0.02 0.04 Observations 44,283 25,576 19,365 11,969 The sample in this table consists of matched pairs of bonds from the same issuer, with yield spreads measured in the same month. Paired bonds mature within 90 days of each other to minimize differential maturity effects. The dependent variable is the yield spread of the longer maturity bond minus the yield spread of the shorter maturity bond. The independent variables are similarly differences in variables, where Prop due prior, T, Rating, Amihud, IRC, Amihud vol, and IRC vol are defined in Table 5. In the first two columns, we match only on maturity (within firm-month). In Columns (3) and (4), we also restrict the matched bonds to have similar Amihud and IRC. Regressions include time fixed effects. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Overall, our results are consistent with a bond’s place in its issuer’s maturity structure affecting its price, even after controlling for time-invariant firm effects, time-varying credit quality proxies, illiquidity variables, the actual maturity of a bond, and the average maturity of bonds issued by the issuer. The results continue to hold in matched sample analysis that focuses on bonds with similar maturities. Our results are consistent with the market understanding the impact of relative maturity on credit risk and suggest that rating agencies should consider de facto seniority in addition to explicit priority structure when notching bonds. 3.2 Hedge ratios Next, we turn to hedge ratios. Whereas yield spreads measure credit quality using the levels of prices, hedge ratios are measured using the joint dynamics of corporate bond and equity returns. Hence, hedge ratios provide a useful alternative test of the effect of de facto seniority. Schaefer and Strebulaev (2008) theoretically show that even if the level of bond prices is different than in a hypothetical frictionless world, models that do not have frictions may match hedge ratios correctly if they provide a reasonable measure of credit risk. The intuition is that although there might be a component in the level of prices related to illiquidity and hence, in returns to corporate bonds due to changes in illiquidity, a regression of bond returns on equity returns will give a coefficient that is largely determined by the covariance of the credit component of corporate bond returns with equity returns. Empirically, Schaefer and Strebulaev (2008) find that hedge ratios are in line with those predicted by the Merton model. For most of this section, we follow the Schaefer and Strebulaev (2008) argument and focus on credit-related variables, but later consider robustness to illiquidity controls. The standard estimate of hedge ratios are from a regression of corporate bond returns on equity returns and Treasury returns,27 \begin{align} r_D = \beta_0 + \beta_E r_E + \beta_T r_T + \varepsilon. \end{align}(9) Intuitively, bonds that have very little credit risk will move very closely with a Treasury of comparable maturity but very little with equities issued by the same firm. The equity hedge ratio estimate, |$\beta_E$|⁠, would be relatively low, whereas the Treasury hedge ratio estimate, |$\beta_T$|⁠, would be relatively high. We estimate hedge ratios in a pooled regression, but allow hedge ratios to vary with different firm and bond characteristics. Before considering the effect of maturity structure, we consider a baseline regression of \begin{align} r_D = \beta_0 + \beta_E r_E + \beta_T r_T + \beta_{E,rat} r_E \times \text{Rating} + \beta_{T,rat} r_T \times \text{Rating} + \beta_{rat} \text{Rating} + \varepsilon. \end{align}(10) Results are reported in panel A of Table 7. The estimated equity hedge ratio of any bond is given by |$\beta_E + \beta_{E,rat}\times\text{Rating}$| and the estimated Treasury hedge ratio is given by |$\beta_T + \beta_{T,rat}\times\text{Rating}$|⁠. In Column (1), we report the estimates of this regression. For a Aaa-rated bond, the regression estimates imply an equity hedge ratio of |$0.0153 + 0.0075\times 1 = 0.0229$| (⁠|$t$|-stat = 1.34) and a Treasury hedge ratio of 0.9450 (⁠|$t$|-stat = 15.03), which are reported in panel B of Table 6. As the regression coefficient on |$r_E\times\text{Rating}$| is positive and the coefficient on |$r_T\times\text{Rating}$| is negative, equity hedge ratios increase and Treasury hedge ratios decrease as ratings decline (numerically coded rating increases). In panel B, we also report the regression-implied hedge ratios for other ratings, along with |$t$|-statistics. We find that starting with an Aa rating, estimated equity hedge ratios begin to be statistically significant. Treasury hedge ratios decline precipitously as ratings decline, all the way to a negative estimated value for B-rated bonds. In Columns (2) and (3) of panel A, we restrict our baseline regression to only investment-grade and speculative-grade bonds, respectively. One difference of note is that although equity hedge ratios increase as credit quality declines, we find that the coefficient on |$r_E\times\text{Rating}$| is insignificant for the speculative-only sample, suggesting that hedge ratios are relatively flat across ratings for speculative-grade bonds. Table 7 Hedge ratios, baseline A. Baseline hedge ratio regressions . . (1) . (2) . (3) . . Full . IG . Junk . |$r_E $| 0.0153 0.0106 0.0551 [0.87] [0.70] [1.08] |$r_T $| 1.0142 0.9133 0.7393 [14.96] [15.53] [3.97] |$r_E \times \text{Rating} $| 0.0075 0.0082 0.0045 [7.39] [3.46] [1.37] |$r_T \times \text{Rating} $| –0.0692 –0.0538 –0.0572 [–9.21] [–7.46] [–4.04] Rating 0.0005 0.0003 0.0005 [4.16] [2.46] [2.32] Constant –0.0006 0.0005 0.0001 [–0.81] [0.93] [0.04] |$R^2$| 0.2440 0.2650 0.1845 Observations 239,837 208,294 31,543 A. Baseline hedge ratio regressions . . (1) . (2) . (3) . . Full . IG . Junk . |$r_E $| 0.0153 0.0106 0.0551 [0.87] [0.70] [1.08] |$r_T $| 1.0142 0.9133 0.7393 [14.96] [15.53] [3.97] |$r_E \times \text{Rating} $| 0.0075 0.0082 0.0045 [7.39] [3.46] [1.37] |$r_T \times \text{Rating} $| –0.0692 –0.0538 –0.0572 [–9.21] [–7.46] [–4.04] Rating 0.0005 0.0003 0.0005 [4.16] [2.46] [2.32] Constant –0.0006 0.0005 0.0001 [–0.81] [0.93] [0.04] |$R^2$| 0.2440 0.2650 0.1845 Observations 239,837 208,294 31,543 B. Estimated hedge ratios Rating Aaa Aa A Baa Ba B Column (1) |$h_E$| 0.0229 0.0380 0.0606 0.0832 0.1058 0.1284 t-stat 1.34 2.38 4.11 5.89 7.48 8.67 |$h_T$| 0.9450 0.8067 0.5992 0.3917 0.1841 –0.0234 t-stat 15.03 14.71 12.14 7.30 2.80 –0.28 Columns (2) & (3) |$h_E$| 0.0188 0.0352 0.0598 0.0844 0.1088 0.1222 t-stat 1.35 2.84 4.50 4.86 7.08 10.94 |$h_T$| 0.8595 0.7520 0.5907 0.4295 0.0530 –0.1186 t-stat 15.67 15.31 12.31 7.68 0.73 –1.43 B. Estimated hedge ratios Rating Aaa Aa A Baa Ba B Column (1) |$h_E$| 0.0229 0.0380 0.0606 0.0832 0.1058 0.1284 t-stat 1.34 2.38 4.11 5.89 7.48 8.67 |$h_T$| 0.9450 0.8067 0.5992 0.3917 0.1841 –0.0234 t-stat 15.03 14.71 12.14 7.30 2.80 –0.28 Columns (2) & (3) |$h_E$| 0.0188 0.0352 0.0598 0.0844 0.1088 0.1222 t-stat 1.35 2.84 4.50 4.86 7.08 10.94 |$h_T$| 0.8595 0.7520 0.5907 0.4295 0.0530 –0.1186 t-stat 15.67 15.31 12.31 7.68 0.73 –1.43 Panel A reports regressions where the dependent variable is log corporate bond returns in decimals. |$r_E$| is the log equity return of the same issuer as the corporate bond and |$r_T$| is the log Treasury bond return of a Treasury bond with the same maturity as the corporate bond. Both |$r_E$| and |$r_T$| are expressed in decimals. Rating is Moody’s rating coded as Aaa=1 and C=21, with intermediate ratings coded. Panel B uses the estimated coefficients in panel A to estimate hedge ratios. In particular, |$h_E = \hat{\beta}_E + \hat{\beta}_{E,rat}\times\text{Rating}$|⁠, where |$\beta_E$| is the coefficient on |$r_E$| and |$\beta_{E,rat}$| is the coefficient on |$r_E\times\text{Rating}$|⁠. The standard error of |$h_E$| uses the variance-covariance matrix from the regressions in panel A and is calculated as |$\text{SD}(\hat{\beta}_E + \hat{\beta}_{E,rat}\times\text{Rating})$|⁠. Similarly, |$h_T = \hat{\beta}_T + \hat{\beta}_{T,rat}\times\text{Rating}$|⁠. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Table 7 Hedge ratios, baseline A. Baseline hedge ratio regressions . . (1) . (2) . (3) . . Full . IG . Junk . |$r_E $| 0.0153 0.0106 0.0551 [0.87] [0.70] [1.08] |$r_T $| 1.0142 0.9133 0.7393 [14.96] [15.53] [3.97] |$r_E \times \text{Rating} $| 0.0075 0.0082 0.0045 [7.39] [3.46] [1.37] |$r_T \times \text{Rating} $| –0.0692 –0.0538 –0.0572 [–9.21] [–7.46] [–4.04] Rating 0.0005 0.0003 0.0005 [4.16] [2.46] [2.32] Constant –0.0006 0.0005 0.0001 [–0.81] [0.93] [0.04] |$R^2$| 0.2440 0.2650 0.1845 Observations 239,837 208,294 31,543 A. Baseline hedge ratio regressions . . (1) . (2) . (3) . . Full . IG . Junk . |$r_E $| 0.0153 0.0106 0.0551 [0.87] [0.70] [1.08] |$r_T $| 1.0142 0.9133 0.7393 [14.96] [15.53] [3.97] |$r_E \times \text{Rating} $| 0.0075 0.0082 0.0045 [7.39] [3.46] [1.37] |$r_T \times \text{Rating} $| –0.0692 –0.0538 –0.0572 [–9.21] [–7.46] [–4.04] Rating 0.0005 0.0003 0.0005 [4.16] [2.46] [2.32] Constant –0.0006 0.0005 0.0001 [–0.81] [0.93] [0.04] |$R^2$| 0.2440 0.2650 0.1845 Observations 239,837 208,294 31,543 B. Estimated hedge ratios Rating Aaa Aa A Baa Ba B Column (1) |$h_E$| 0.0229 0.0380 0.0606 0.0832 0.1058 0.1284 t-stat 1.34 2.38 4.11 5.89 7.48 8.67 |$h_T$| 0.9450 0.8067 0.5992 0.3917 0.1841 –0.0234 t-stat 15.03 14.71 12.14 7.30 2.80 –0.28 Columns (2) & (3) |$h_E$| 0.0188 0.0352 0.0598 0.0844 0.1088 0.1222 t-stat 1.35 2.84 4.50 4.86 7.08 10.94 |$h_T$| 0.8595 0.7520 0.5907 0.4295 0.0530 –0.1186 t-stat 15.67 15.31 12.31 7.68 0.73 –1.43 B. Estimated hedge ratios Rating Aaa Aa A Baa Ba B Column (1) |$h_E$| 0.0229 0.0380 0.0606 0.0832 0.1058 0.1284 t-stat 1.34 2.38 4.11 5.89 7.48 8.67 |$h_T$| 0.9450 0.8067 0.5992 0.3917 0.1841 –0.0234 t-stat 15.03 14.71 12.14 7.30 2.80 –0.28 Columns (2) & (3) |$h_E$| 0.0188 0.0352 0.0598 0.0844 0.1088 0.1222 t-stat 1.35 2.84 4.50 4.86 7.08 10.94 |$h_T$| 0.8595 0.7520 0.5907 0.4295 0.0530 –0.1186 t-stat 15.67 15.31 12.31 7.68 0.73 –1.43 Panel A reports regressions where the dependent variable is log corporate bond returns in decimals. |$r_E$| is the log equity return of the same issuer as the corporate bond and |$r_T$| is the log Treasury bond return of a Treasury bond with the same maturity as the corporate bond. Both |$r_E$| and |$r_T$| are expressed in decimals. Rating is Moody’s rating coded as Aaa=1 and C=21, with intermediate ratings coded. Panel B uses the estimated coefficients in panel A to estimate hedge ratios. In particular, |$h_E = \hat{\beta}_E + \hat{\beta}_{E,rat}\times\text{Rating}$|⁠, where |$\beta_E$| is the coefficient on |$r_E$| and |$\beta_{E,rat}$| is the coefficient on |$r_E\times\text{Rating}$|⁠. The standard error of |$h_E$| uses the variance-covariance matrix from the regressions in panel A and is calculated as |$\text{SD}(\hat{\beta}_E + \hat{\beta}_{E,rat}\times\text{Rating})$|⁠. Similarly, |$h_T = \hat{\beta}_T + \hat{\beta}_{T,rat}\times\text{Rating}$|⁠. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab To study the de facto seniority effect, we introduce an interaction term between equity returns and proportion due prior to the regression and report the results in Table 8. The estimate of a particular bond’s equity hedge ratio is now Table 8 Hedge ratios A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . . Full . Full . Full . IG . Junk . No crisis . |$r_E \times \text{Prop due prior}$| 0.0791 0.0716 0.0861 0.0829 0.0994 0.0693 [5.97] [5.74] [5.28] [4.06] [5.21] [5.95] |$r_E $| –0.0385 –0.0492 –0.0693 –0.0714 –0.0297 –0.0863 [–3.22] [–3.59] [–5.48] [–3.79] [–0.51] [–6.20] |$r_T $| 1.0112 1.0161 1.0143 0.9124 0.7932 0.9688 [15.01] [15.07] [15.04] [15.30] [3.97] [17.85] |$r_E \times \text{T} $| 0.0015 0.0018 0.0012 0.0017 –0.0008 0.0017 [2.62] [3.42] [3.10] [4.38] [–0.83] [4.10] |$r_E \times \text{Rating} $| 0.0085 0.0059 0.0061 0.0051 0.0046 0.0044 [8.13] [4.23] [4.23] [2.20] [1.19] [4.01] |$r_E \times \text{BE/ME} $| 0.0314 0.0313 0.0427 0.0192 0.0578 [2.24] [2.25] [2.78] [1.23] [6.20] |$r_E \times \text{Equity vol} $| 0.0274 0.0307 0.0257 0.0361 0.0454 [0.94] [1.05] [0.74] [0.92] [1.30] |$r_E \times \text{Firm T} $| 0.0016 0.0017 0.0016 0.0016 [1.78] [1.91] [1.22] [1.85] |$r_T \times \text{Rating} $| –0.0671 –0.0676 –0.0673 –0.0516 –0.0610 –0.0513 [–8.81] [–8.60] [–8.52] [–7.05] [–3.89] [–7.12] Prop due prior 0.0012 0.0003 0.0009 0.0012 0.0002 0.0016 [1.57] [0.50] [0.94] [1.16] [0.11] [2.01] T –0.0000 0.0000 –0.0000 –0.0000 0.0000 –0.0000 [–0.10] [0.34] [–0.25] [–0.58] [0.73] [–0.47] Rating 0.0005 –0.0000 –0.0000 –0.0000 –0.0001 0.0002 [4.08] [–0.16] [–0.13] [–0.04] [–0.23] [1.71] BE/ME 0.0022 0.0021 0.0016 0.0030 0.0006 [2.19] [2.12] [1.39] [1.99] [0.93] Equity vol 0.0248 0.0248 0.0251 0.0241 0.0174 [4.43] [4.42] [4.31] [3.51] [4.96] Firm T 0.0001 0.0001 0.0001 0.0000 [1.14] [1.06] [1.08] [1.39] Constant –0.0011 –0.0054 –0.0061 –0.0060 –0.0062 –0.0046 [–1.56] [–3.97] [–3.60] [–3.07] [–2.00] [–4.14] |$R^2$| 0.2573 0.2845 0.2850 0.3076 0.2218 0.3604 Observations 239,837 233,706 233,706 205,055 28,651 194,822 A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . . Full . Full . Full . IG . Junk . No crisis . |$r_E \times \text{Prop due prior}$| 0.0791 0.0716 0.0861 0.0829 0.0994 0.0693 [5.97] [5.74] [5.28] [4.06] [5.21] [5.95] |$r_E $| –0.0385 –0.0492 –0.0693 –0.0714 –0.0297 –0.0863 [–3.22] [–3.59] [–5.48] [–3.79] [–0.51] [–6.20] |$r_T $| 1.0112 1.0161 1.0143 0.9124 0.7932 0.9688 [15.01] [15.07] [15.04] [15.30] [3.97] [17.85] |$r_E \times \text{T} $| 0.0015 0.0018 0.0012 0.0017 –0.0008 0.0017 [2.62] [3.42] [3.10] [4.38] [–0.83] [4.10] |$r_E \times \text{Rating} $| 0.0085 0.0059 0.0061 0.0051 0.0046 0.0044 [8.13] [4.23] [4.23] [2.20] [1.19] [4.01] |$r_E \times \text{BE/ME} $| 0.0314 0.0313 0.0427 0.0192 0.0578 [2.24] [2.25] [2.78] [1.23] [6.20] |$r_E \times \text{Equity vol} $| 0.0274 0.0307 0.0257 0.0361 0.0454 [0.94] [1.05] [0.74] [0.92] [1.30] |$r_E \times \text{Firm T} $| 0.0016 0.0017 0.0016 0.0016 [1.78] [1.91] [1.22] [1.85] |$r_T \times \text{Rating} $| –0.0671 –0.0676 –0.0673 –0.0516 –0.0610 –0.0513 [–8.81] [–8.60] [–8.52] [–7.05] [–3.89] [–7.12] Prop due prior 0.0012 0.0003 0.0009 0.0012 0.0002 0.0016 [1.57] [0.50] [0.94] [1.16] [0.11] [2.01] T –0.0000 0.0000 –0.0000 –0.0000 0.0000 –0.0000 [–0.10] [0.34] [–0.25] [–0.58] [0.73] [–0.47] Rating 0.0005 –0.0000 –0.0000 –0.0000 –0.0001 0.0002 [4.08] [–0.16] [–0.13] [–0.04] [–0.23] [1.71] BE/ME 0.0022 0.0021 0.0016 0.0030 0.0006 [2.19] [2.12] [1.39] [1.99] [0.93] Equity vol 0.0248 0.0248 0.0251 0.0241 0.0174 [4.43] [4.42] [4.31] [3.51] [4.96] Firm T 0.0001 0.0001 0.0001 0.0000 [1.14] [1.06] [1.08] [1.39] Constant –0.0011 –0.0054 –0.0061 –0.0060 –0.0062 –0.0046 [–1.56] [–3.97] [–3.60] [–3.07] [–2.00] [–4.14] |$R^2$| 0.2573 0.2845 0.2850 0.3076 0.2218 0.3604 Observations 239,837 233,706 233,706 205,055 28,651 194,822 B. With firm fixed effects (1) (2) (3) (4) (5) (6) Full Full Full IG Junk No crisis |$r_E \times \text{Prop due prior} $| 0.0732 0.0721 0.0852 0.0856 0.0881 0.0742 [4.73] [4.69] [4.59] [4.22] [4.01] [6.44] |$r_T $| 1.0109 1.0142 1.0144 0.9116 0.8100 0.9710 [15.15] [15.50] [15.51] [15.52] [4.00] [17.90] |$r_E \times \text{T} $| 0.0018 0.0019 0.0015 0.0017 0.0006 0.0019 [5.34] [5.40] [4.33] [4.10] [1.06] [4.52] |$r_E \times \text{Rating} $| –0.0002 –0.0007 –0.0004 0.0004 –0.0039 0.0017 [–0.08] [–0.29] [–0.18] [0.15] [–0.70] [0.52] |$r_E \times \text{BE/ME} $| 0.0269 0.0283 0.0364 0.0253 0.0425 [1.51] [1.65] [1.46] [1.64] [2.96] |$r_E \times \text{Equity vol} $| 0.0013 0.0006 –0.0141 0.0124 0.0084 [0.04] [0.02] [–0.36] [0.28] [0.21] |$r_E \times \text{Firm T} $| 0.0063 0.0056 0.0092 0.0044 [2.73] [2.80] [1.80] [3.19] |$r_T \times \text{Rating} $| –0.0671 –0.0670 –0.0670 –0.0511 –0.0622 –0.0517 [–8.91] [–8.67] [–8.67] [–7.24] [–3.89] [–7.32] Prop due prior 0.0016 0.0012 0.0011 0.0012 0.0015 0.0016 [1.62] [1.23] [1.02] [1.11] [0.89] [1.78] T –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 [–0.75] [–0.43] [–0.39] [–0.52] [–0.13] [–0.64] Rating 0.0010 0.0004 0.0004 0.0002 0.0002 0.0005 [5.72] [2.18] [2.12] [1.41] [0.54] [2.68] BE/ME 0.0049 0.0048 0.0047 0.0051 0.0025 [2.13] [2.10] [1.92] [1.71] [1.73] Equity vol 0.0312 0.0313 0.0323 0.0279 0.0224 [4.61] [4.62] [4.44] [3.45] [4.78] Firm T –0.0000 –0.0001 0.0001 –0.0000 [–0.50] [–0.65] [0.16] [–0.33] |$R^2$| 0.2877 0.3167 0.3176 0.3363 0.2784 0.3946 Observations 239,837 233,706 233,706 205,055 28,651 194,822 B. With firm fixed effects (1) (2) (3) (4) (5) (6) Full Full Full IG Junk No crisis |$r_E \times \text{Prop due prior} $| 0.0732 0.0721 0.0852 0.0856 0.0881 0.0742 [4.73] [4.69] [4.59] [4.22] [4.01] [6.44] |$r_T $| 1.0109 1.0142 1.0144 0.9116 0.8100 0.9710 [15.15] [15.50] [15.51] [15.52] [4.00] [17.90] |$r_E \times \text{T} $| 0.0018 0.0019 0.0015 0.0017 0.0006 0.0019 [5.34] [5.40] [4.33] [4.10] [1.06] [4.52] |$r_E \times \text{Rating} $| –0.0002 –0.0007 –0.0004 0.0004 –0.0039 0.0017 [–0.08] [–0.29] [–0.18] [0.15] [–0.70] [0.52] |$r_E \times \text{BE/ME} $| 0.0269 0.0283 0.0364 0.0253 0.0425 [1.51] [1.65] [1.46] [1.64] [2.96] |$r_E \times \text{Equity vol} $| 0.0013 0.0006 –0.0141 0.0124 0.0084 [0.04] [0.02] [–0.36] [0.28] [0.21] |$r_E \times \text{Firm T} $| 0.0063 0.0056 0.0092 0.0044 [2.73] [2.80] [1.80] [3.19] |$r_T \times \text{Rating} $| –0.0671 –0.0670 –0.0670 –0.0511 –0.0622 –0.0517 [–8.91] [–8.67] [–8.67] [–7.24] [–3.89] [–7.32] Prop due prior 0.0016 0.0012 0.0011 0.0012 0.0015 0.0016 [1.62] [1.23] [1.02] [1.11] [0.89] [1.78] T –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 [–0.75] [–0.43] [–0.39] [–0.52] [–0.13] [–0.64] Rating 0.0010 0.0004 0.0004 0.0002 0.0002 0.0005 [5.72] [2.18] [2.12] [1.41] [0.54] [2.68] BE/ME 0.0049 0.0048 0.0047 0.0051 0.0025 [2.13] [2.10] [1.92] [1.71] [1.73] Equity vol 0.0312 0.0313 0.0323 0.0279 0.0224 [4.61] [4.62] [4.44] [3.45] [4.78] Firm T –0.0000 –0.0001 0.0001 –0.0000 [–0.50] [–0.65] [0.16] [–0.33] |$R^2$| 0.2877 0.3167 0.3176 0.3363 0.2784 0.3946 Observations 239,837 233,706 233,706 205,055 28,651 194,822 The dependent variable in this table is log corporate bond returns in decimals. |$r_E$| is the log equity return of the same issuer as the corporate bond and |$r_T$| is the log Treasury bond return of a Treasury bond with the same maturity as the corporate bond. Both |$r_E$| and |$r_T$| are expressed in decimals. Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. Rating is Moody’s rating coded as Aaa=1 and C=21, with intermediate ratings coded. BE/ME is book equity divided by market equity for the bond issuer, expressed in decimals. Equity vol is an annualized equity volatility calculated from daily returns in the last year in decimals. Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Table 8 Hedge ratios A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . . Full . Full . Full . IG . Junk . No crisis . |$r_E \times \text{Prop due prior}$| 0.0791 0.0716 0.0861 0.0829 0.0994 0.0693 [5.97] [5.74] [5.28] [4.06] [5.21] [5.95] |$r_E $| –0.0385 –0.0492 –0.0693 –0.0714 –0.0297 –0.0863 [–3.22] [–3.59] [–5.48] [–3.79] [–0.51] [–6.20] |$r_T $| 1.0112 1.0161 1.0143 0.9124 0.7932 0.9688 [15.01] [15.07] [15.04] [15.30] [3.97] [17.85] |$r_E \times \text{T} $| 0.0015 0.0018 0.0012 0.0017 –0.0008 0.0017 [2.62] [3.42] [3.10] [4.38] [–0.83] [4.10] |$r_E \times \text{Rating} $| 0.0085 0.0059 0.0061 0.0051 0.0046 0.0044 [8.13] [4.23] [4.23] [2.20] [1.19] [4.01] |$r_E \times \text{BE/ME} $| 0.0314 0.0313 0.0427 0.0192 0.0578 [2.24] [2.25] [2.78] [1.23] [6.20] |$r_E \times \text{Equity vol} $| 0.0274 0.0307 0.0257 0.0361 0.0454 [0.94] [1.05] [0.74] [0.92] [1.30] |$r_E \times \text{Firm T} $| 0.0016 0.0017 0.0016 0.0016 [1.78] [1.91] [1.22] [1.85] |$r_T \times \text{Rating} $| –0.0671 –0.0676 –0.0673 –0.0516 –0.0610 –0.0513 [–8.81] [–8.60] [–8.52] [–7.05] [–3.89] [–7.12] Prop due prior 0.0012 0.0003 0.0009 0.0012 0.0002 0.0016 [1.57] [0.50] [0.94] [1.16] [0.11] [2.01] T –0.0000 0.0000 –0.0000 –0.0000 0.0000 –0.0000 [–0.10] [0.34] [–0.25] [–0.58] [0.73] [–0.47] Rating 0.0005 –0.0000 –0.0000 –0.0000 –0.0001 0.0002 [4.08] [–0.16] [–0.13] [–0.04] [–0.23] [1.71] BE/ME 0.0022 0.0021 0.0016 0.0030 0.0006 [2.19] [2.12] [1.39] [1.99] [0.93] Equity vol 0.0248 0.0248 0.0251 0.0241 0.0174 [4.43] [4.42] [4.31] [3.51] [4.96] Firm T 0.0001 0.0001 0.0001 0.0000 [1.14] [1.06] [1.08] [1.39] Constant –0.0011 –0.0054 –0.0061 –0.0060 –0.0062 –0.0046 [–1.56] [–3.97] [–3.60] [–3.07] [–2.00] [–4.14] |$R^2$| 0.2573 0.2845 0.2850 0.3076 0.2218 0.3604 Observations 239,837 233,706 233,706 205,055 28,651 194,822 A. Without firm fixed effects . . (1) . (2) . (3) . (4) . (5) . (6) . . Full . Full . Full . IG . Junk . No crisis . |$r_E \times \text{Prop due prior}$| 0.0791 0.0716 0.0861 0.0829 0.0994 0.0693 [5.97] [5.74] [5.28] [4.06] [5.21] [5.95] |$r_E $| –0.0385 –0.0492 –0.0693 –0.0714 –0.0297 –0.0863 [–3.22] [–3.59] [–5.48] [–3.79] [–0.51] [–6.20] |$r_T $| 1.0112 1.0161 1.0143 0.9124 0.7932 0.9688 [15.01] [15.07] [15.04] [15.30] [3.97] [17.85] |$r_E \times \text{T} $| 0.0015 0.0018 0.0012 0.0017 –0.0008 0.0017 [2.62] [3.42] [3.10] [4.38] [–0.83] [4.10] |$r_E \times \text{Rating} $| 0.0085 0.0059 0.0061 0.0051 0.0046 0.0044 [8.13] [4.23] [4.23] [2.20] [1.19] [4.01] |$r_E \times \text{BE/ME} $| 0.0314 0.0313 0.0427 0.0192 0.0578 [2.24] [2.25] [2.78] [1.23] [6.20] |$r_E \times \text{Equity vol} $| 0.0274 0.0307 0.0257 0.0361 0.0454 [0.94] [1.05] [0.74] [0.92] [1.30] |$r_E \times \text{Firm T} $| 0.0016 0.0017 0.0016 0.0016 [1.78] [1.91] [1.22] [1.85] |$r_T \times \text{Rating} $| –0.0671 –0.0676 –0.0673 –0.0516 –0.0610 –0.0513 [–8.81] [–8.60] [–8.52] [–7.05] [–3.89] [–7.12] Prop due prior 0.0012 0.0003 0.0009 0.0012 0.0002 0.0016 [1.57] [0.50] [0.94] [1.16] [0.11] [2.01] T –0.0000 0.0000 –0.0000 –0.0000 0.0000 –0.0000 [–0.10] [0.34] [–0.25] [–0.58] [0.73] [–0.47] Rating 0.0005 –0.0000 –0.0000 –0.0000 –0.0001 0.0002 [4.08] [–0.16] [–0.13] [–0.04] [–0.23] [1.71] BE/ME 0.0022 0.0021 0.0016 0.0030 0.0006 [2.19] [2.12] [1.39] [1.99] [0.93] Equity vol 0.0248 0.0248 0.0251 0.0241 0.0174 [4.43] [4.42] [4.31] [3.51] [4.96] Firm T 0.0001 0.0001 0.0001 0.0000 [1.14] [1.06] [1.08] [1.39] Constant –0.0011 –0.0054 –0.0061 –0.0060 –0.0062 –0.0046 [–1.56] [–3.97] [–3.60] [–3.07] [–2.00] [–4.14] |$R^2$| 0.2573 0.2845 0.2850 0.3076 0.2218 0.3604 Observations 239,837 233,706 233,706 205,055 28,651 194,822 B. With firm fixed effects (1) (2) (3) (4) (5) (6) Full Full Full IG Junk No crisis |$r_E \times \text{Prop due prior} $| 0.0732 0.0721 0.0852 0.0856 0.0881 0.0742 [4.73] [4.69] [4.59] [4.22] [4.01] [6.44] |$r_T $| 1.0109 1.0142 1.0144 0.9116 0.8100 0.9710 [15.15] [15.50] [15.51] [15.52] [4.00] [17.90] |$r_E \times \text{T} $| 0.0018 0.0019 0.0015 0.0017 0.0006 0.0019 [5.34] [5.40] [4.33] [4.10] [1.06] [4.52] |$r_E \times \text{Rating} $| –0.0002 –0.0007 –0.0004 0.0004 –0.0039 0.0017 [–0.08] [–0.29] [–0.18] [0.15] [–0.70] [0.52] |$r_E \times \text{BE/ME} $| 0.0269 0.0283 0.0364 0.0253 0.0425 [1.51] [1.65] [1.46] [1.64] [2.96] |$r_E \times \text{Equity vol} $| 0.0013 0.0006 –0.0141 0.0124 0.0084 [0.04] [0.02] [–0.36] [0.28] [0.21] |$r_E \times \text{Firm T} $| 0.0063 0.0056 0.0092 0.0044 [2.73] [2.80] [1.80] [3.19] |$r_T \times \text{Rating} $| –0.0671 –0.0670 –0.0670 –0.0511 –0.0622 –0.0517 [–8.91] [–8.67] [–8.67] [–7.24] [–3.89] [–7.32] Prop due prior 0.0016 0.0012 0.0011 0.0012 0.0015 0.0016 [1.62] [1.23] [1.02] [1.11] [0.89] [1.78] T –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 [–0.75] [–0.43] [–0.39] [–0.52] [–0.13] [–0.64] Rating 0.0010 0.0004 0.0004 0.0002 0.0002 0.0005 [5.72] [2.18] [2.12] [1.41] [0.54] [2.68] BE/ME 0.0049 0.0048 0.0047 0.0051 0.0025 [2.13] [2.10] [1.92] [1.71] [1.73] Equity vol 0.0312 0.0313 0.0323 0.0279 0.0224 [4.61] [4.62] [4.44] [3.45] [4.78] Firm T –0.0000 –0.0001 0.0001 –0.0000 [–0.50] [–0.65] [0.16] [–0.33] |$R^2$| 0.2877 0.3167 0.3176 0.3363 0.2784 0.3946 Observations 239,837 233,706 233,706 205,055 28,651 194,822 B. With firm fixed effects (1) (2) (3) (4) (5) (6) Full Full Full IG Junk No crisis |$r_E \times \text{Prop due prior} $| 0.0732 0.0721 0.0852 0.0856 0.0881 0.0742 [4.73] [4.69] [4.59] [4.22] [4.01] [6.44] |$r_T $| 1.0109 1.0142 1.0144 0.9116 0.8100 0.9710 [15.15] [15.50] [15.51] [15.52] [4.00] [17.90] |$r_E \times \text{T} $| 0.0018 0.0019 0.0015 0.0017 0.0006 0.0019 [5.34] [5.40] [4.33] [4.10] [1.06] [4.52] |$r_E \times \text{Rating} $| –0.0002 –0.0007 –0.0004 0.0004 –0.0039 0.0017 [–0.08] [–0.29] [–0.18] [0.15] [–0.70] [0.52] |$r_E \times \text{BE/ME} $| 0.0269 0.0283 0.0364 0.0253 0.0425 [1.51] [1.65] [1.46] [1.64] [2.96] |$r_E \times \text{Equity vol} $| 0.0013 0.0006 –0.0141 0.0124 0.0084 [0.04] [0.02] [–0.36] [0.28] [0.21] |$r_E \times \text{Firm T} $| 0.0063 0.0056 0.0092 0.0044 [2.73] [2.80] [1.80] [3.19] |$r_T \times \text{Rating} $| –0.0671 –0.0670 –0.0670 –0.0511 –0.0622 –0.0517 [–8.91] [–8.67] [–8.67] [–7.24] [–3.89] [–7.32] Prop due prior 0.0016 0.0012 0.0011 0.0012 0.0015 0.0016 [1.62] [1.23] [1.02] [1.11] [0.89] [1.78] T –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 –0.0000 [–0.75] [–0.43] [–0.39] [–0.52] [–0.13] [–0.64] Rating 0.0010 0.0004 0.0004 0.0002 0.0002 0.0005 [5.72] [2.18] [2.12] [1.41] [0.54] [2.68] BE/ME 0.0049 0.0048 0.0047 0.0051 0.0025 [2.13] [2.10] [1.92] [1.71] [1.73] Equity vol 0.0312 0.0313 0.0323 0.0279 0.0224 [4.61] [4.62] [4.44] [3.45] [4.78] Firm T –0.0000 –0.0001 0.0001 –0.0000 [–0.50] [–0.65] [0.16] [–0.33] |$R^2$| 0.2877 0.3167 0.3176 0.3363 0.2784 0.3946 Observations 239,837 233,706 233,706 205,055 28,651 194,822 The dependent variable in this table is log corporate bond returns in decimals. |$r_E$| is the log equity return of the same issuer as the corporate bond and |$r_T$| is the log Treasury bond return of a Treasury bond with the same maturity as the corporate bond. Both |$r_E$| and |$r_T$| are expressed in decimals. Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. Rating is Moody’s rating coded as Aaa=1 and C=21, with intermediate ratings coded. BE/ME is book equity divided by market equity for the bond issuer, expressed in decimals. Equity vol is an annualized equity volatility calculated from daily returns in the last year in decimals. Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab \begin{align} \beta_E + \sum_i(\text{coefficient on}r_E\times\text{char}_i)\times\text{char}_i, \end{align}(11) where |$\text{char}_i$| is a bond or firm characteristic, including proportion due prior. The coefficient on |$r_E\times\text{proportion due prior}$| measures the marginal effect of proportion due prior on the equity hedge ratio. We also include interaction terms between equity returns and bond maturity (T), equity book-to-market (BE/ME), equity volatility, and average firm-level bond maturity of the issuer (Firm T) in addition to the interactions with ratings and proportion due prior.28 We find an economically and statistically significant effect for proportion due prior on hedge ratios. Controlling for the effects of ratings, book-to-market, equity volatility, bond maturity, and firm-level maturity on hedge ratios (Column (3)), we find that a bond that is the latest bond in its issuer’s maturity structure has an equity hedge ratio that is 0.0861 larger than the earliest bond in its issuer’s maturity structure.29 Compared to estimated hedge ratios of 0.0606 and 0.0832 for A and Baa bonds (which represent 78% of our bond-month observations) in Table 7, respectively, this is an economically large effect. Running regressions for only investment-grade (Column (4)) and only speculative-grade (Column (5)) bonds, we continue to find a significant effect of proportion due prior on equity hedge ratios. However, we find that the effect for speculative-grade bonds is not much higher than for investment-grade bonds.30 Focusing on only noncrisis periods (omitting July 2007 to June 2009) in specification (6), we find a marginally lower, but still economically and statistically significant coefficient on |$r_E\times\text{proportion due prior}$|⁠, suggesting that our results are not driven solely by the financial crisis. In panel B of Table 8, we introduce interaction terms between equity returns and firm fixed effects along with the fixed effects themselves. This allows for a firm-specific effect in equity hedge ratios. We find that this has little impact on the marginal effect of proportion due prior on hedge ratios. A bond that is latest in its firm’s maturity structure has a hedge ratio that is larger than the earliest bond by 0.0852 (Column (3)). Throughout this subsection, we have left out liquidity variables. This is because Schaefer and Strebulaev (2008) theoretically and empirically show that the impact of nondefault components of bond prices on hedge ratios is small even even though they have large effects on the levels of prices. Nevertheless, we augment specification (3) of Table 8, panel B with |$r_E\times\text{Amihud}$| and |$r_E\times\text{IRC}$| (along with the corresponding level terms) as a robustness check. We find that the coefficient on |$r_E\times\text{proportion due prior}$| is 0.0752 (⁠|$t$|-stat |$=$| 3.97), as compared to 0.0852 (⁠|$t$|-stat |$=$| 4.59) without liquidity controls. Whereas |$r_E\times\text{Amihud}$| is statistically insignificant, |$r_E\times\text{IRC}$| is significant with a |$t$|-stat of 2.85. A move from the 10th to 90th percentile of IRC (a relatively large move) is associated with a change in hedge ratio of 0.0167, a relatively small effect.31 Thus, our results (similar to the theoretical arguments in Schaefer and Strebulaev 2008), are consistent with illiquidity playing a relatively minor role in hedge ratios. Overall, our results suggest that in addition to yield spreads, there is also a very significant de facto seniority effect in hedge ratios. 3.3 Model versus empirical effect Though the extended Merton model developed in Section 1 is largely designed to illustrate the intuition of how the de facto seniority affects yield spreads and hedge ratios rather than fully reflect the many complexities of firms’ debt structures, here we consider how closely our empirical estimates match the model-implied effects of de facto seniority. Similar to Schaefer and Strebulaev (2008), we are not claiming that simple structural models are precise representations of a firm, but are instead interested in whether such models produce reasonable quantitative approximations. To calculate model-implied de facto seniority effects, we start with the (⁠|$\sigma_v$|⁠, market leverage, proportion due prior) triplet for each observation in our sample. For each observation, we calculate the model-implied yield spread effect as \begin{align} \frac{f(\sigma_v,\text{mkt lev},\text{prop due prior}+a) - f(\sigma_v,\text{mkt lev},\text{prop due prior}-a)}{2a}, \end{align}(12) where f(.) is a function representing the model yield spread as a function of asset volatility, market leverage, and proportion due prior. Essentially, our goal is to calculate the marginal change in yield spreads for a small change in proportion due prior. Similarly, we calculate a model-implied hedge ratio effect. Appendix D provides additional details of our calculations. After obtaining a model-implied de facto seniority effect in yield spreads for each observation in our sample, we estimate a seemingly unrelated regression (SUR), which allows for covariances in coefficient estimates across equations, \begin{align} &\text{Model estimated yield spread effect}_{it} = \zeta_0 + e_{it}\label{eq13}\\ \end{align}(13) \begin{align} &\text{Actual yield spread}_{it} = \alpha_j + \beta_1 \text{prop due prior}_{it} + \beta_2 T_{it} + \beta_3 \text{Rating}_{it} + \beta_4 X_{it} + \varepsilon_{it},\label{eq:sur} \end{align}(14) where |$X_{it}$| is a set of controls. Equation (14) is the same regression from Table 5, but is reestimated together with model-implied yield spread effects to allow for covariances in the parameter estimates across equations. To test the model, we run a Wald test with a null hypothesis that the model-implied effect and empirical effect are equal, |$\zeta_0 = \beta_1$|⁠. In panel A of Table 9, we report our tests using yield spreads. We consider both the full sample and also investment-grade and junk subsamples because, as we will discuss later, there are reasons to believe that our model will fit investment-grade firms better. As shown in rows (1) to (3), the model is consistently rejected when the full sample is used, regardless of what controls are included in the empirical specification. The empirical estimates of the de facto seniority effect are statistically and economically smaller than the model-implied effect. However, in Columns (4) to (6), we see that for the investment-grade subsample, the model cannot be rejected in two out of three specifications. Rows (7) to (9) show that the model is strongly rejected for the junk bond subsample as |$p$|-values are less than 0.01 for all specifications. Thus, despite the fact that empirically the de facto seniority effect is stronger for junk bonds than investment grade bonds, these tests suggest that the relative increase in the de facto seniority effect when going from investment-grade to junk is not nearly large enough as compared to what the extended Merton model predicts. Table 9 Model versus empirical estimates A. Yield spreads . . . Model . Empirical . . Model test . . . Avg . SE . Coeff . SE . Controls . |$\chi^2$| . p . (1) Full 1.1253 0.1678 0.7001 0.0567 T, Rating, time FE, firm FE 5.5384 0.0186 (2) Full 1.0993 0.1679 0.6625 0.0576 (1) + BE/ME, Eq vol, Firm T 5.5009 0.0190 (3) Full 1.1811 0.1733 0.4773 0.0504 (2) + Liquidity 13.4658 0.0002 (4) IG 0.8110 0.1453 0.6448 0.0537 T, Rating, time FE, firm FE 1.3180 0.2509 (5) IG 0.8110 0.1458 0.6229 0.0521 (4) + BE/ME, Eq vol, Firm T 1.5843 0.2081 (6) IG 0.8467 0.1442 0.4670 0.0412 (5) + Liquidity 6.1588 0.0131 (7) Junk 3.1838 0.4155 0.9848 0.1280 T, Rating, time FE, firm FE 20.3059 0.0000 (8) Junk 3.1345 0.4391 1.1060 0.1481 (7) + BE/ME, Eq vol, Firm T 14.6705 0.0001 (9) Junk 3.4480 0.4493 0.8510 0.1292 (8) + Liquidity 25.1677 0.0000 B. Hedge ratios Model Empirical Model test Avg SE Coeff SE Controls (interacted with |$r_E$|⁠) |$\chi^2$| p (1) Full 0.0820 0.0082 0.0732 0.0154 T, Rating, firm FE 0.2601 0.6100 (2) Full 0.0800 0.0081 0.0852 0.0185 (1) + BE/ME, Eq vol, Firm T 0.0660 0.7972 (3) Full 0.0831 0.0082 0.0704 0.0159 (2) + Liquidity 0.5147 0.4731 (4) IG 0.0659 0.0079 0.0735 0.0171 T, Rating, firm FE 0.1792 0.6720 (5) IG 0.0658 0.0080 0.0856 0.0202 (4) + BE/ME, Eq vol, Firm T 0.8850 0.3468 (6) IG 0.0674 0.0080 0.0665 0.0213 (5) + Liquidity 0.0019 0.9657 (7) Junk 0.1886 0.0150 0.0777 0.0179 T, Rating, firm FE 21.1105 0.0000 (8) Junk 0.1818 0.0137 0.0881 0.0218 (7) + BE/ME, Eq vol, Firm T 12.0823 0.0005 (9) Junk 0.1924 0.0137 0.0832 0.0130 (8) + Liquidity 33.3422 0.0000 A. Yield spreads . . . Model . Empirical . . Model test . . . Avg . SE . Coeff . SE . Controls . |$\chi^2$| . p . (1) Full 1.1253 0.1678 0.7001 0.0567 T, Rating, time FE, firm FE 5.5384 0.0186 (2) Full 1.0993 0.1679 0.6625 0.0576 (1) + BE/ME, Eq vol, Firm T 5.5009 0.0190 (3) Full 1.1811 0.1733 0.4773 0.0504 (2) + Liquidity 13.4658 0.0002 (4) IG 0.8110 0.1453 0.6448 0.0537 T, Rating, time FE, firm FE 1.3180 0.2509 (5) IG 0.8110 0.1458 0.6229 0.0521 (4) + BE/ME, Eq vol, Firm T 1.5843 0.2081 (6) IG 0.8467 0.1442 0.4670 0.0412 (5) + Liquidity 6.1588 0.0131 (7) Junk 3.1838 0.4155 0.9848 0.1280 T, Rating, time FE, firm FE 20.3059 0.0000 (8) Junk 3.1345 0.4391 1.1060 0.1481 (7) + BE/ME, Eq vol, Firm T 14.6705 0.0001 (9) Junk 3.4480 0.4493 0.8510 0.1292 (8) + Liquidity 25.1677 0.0000 B. Hedge ratios Model Empirical Model test Avg SE Coeff SE Controls (interacted with |$r_E$|⁠) |$\chi^2$| p (1) Full 0.0820 0.0082 0.0732 0.0154 T, Rating, firm FE 0.2601 0.6100 (2) Full 0.0800 0.0081 0.0852 0.0185 (1) + BE/ME, Eq vol, Firm T 0.0660 0.7972 (3) Full 0.0831 0.0082 0.0704 0.0159 (2) + Liquidity 0.5147 0.4731 (4) IG 0.0659 0.0079 0.0735 0.0171 T, Rating, firm FE 0.1792 0.6720 (5) IG 0.0658 0.0080 0.0856 0.0202 (4) + BE/ME, Eq vol, Firm T 0.8850 0.3468 (6) IG 0.0674 0.0080 0.0665 0.0213 (5) + Liquidity 0.0019 0.9657 (7) Junk 0.1886 0.0150 0.0777 0.0179 T, Rating, firm FE 21.1105 0.0000 (8) Junk 0.1818 0.0137 0.0881 0.0218 (7) + BE/ME, Eq vol, Firm T 12.0823 0.0005 (9) Junk 0.1924 0.0137 0.0832 0.0130 (8) + Liquidity 33.3422 0.0000 This table compares extended Merton model-implied yield spreads and hedge ratio effects with empirically estimated effects. Panel A considers yield spreads and panel B hedge ratios. The model columns display the average effect after calculating observation-by-observation model effects and the standard error is for the average. The empirical columns have estimates that correspond to the coefficient on proportion due prior in the yield spread regressions and |$r_E\times\text{proportion due prior}$| in the hedge ratio regressions. For the hedge ratio regressions, the controls include both interaction terms with |$r_E$| and also the controls on their own. The last two columns present the results of Wald test for which the null hypothesis is that the model and empirical effects are equal. Standard errors are two-way clustered by firm and time. Open in new tab Table 9 Model versus empirical estimates A. Yield spreads . . . Model . Empirical . . Model test . . . Avg . SE . Coeff . SE . Controls . |$\chi^2$| . p . (1) Full 1.1253 0.1678 0.7001 0.0567 T, Rating, time FE, firm FE 5.5384 0.0186 (2) Full 1.0993 0.1679 0.6625 0.0576 (1) + BE/ME, Eq vol, Firm T 5.5009 0.0190 (3) Full 1.1811 0.1733 0.4773 0.0504 (2) + Liquidity 13.4658 0.0002 (4) IG 0.8110 0.1453 0.6448 0.0537 T, Rating, time FE, firm FE 1.3180 0.2509 (5) IG 0.8110 0.1458 0.6229 0.0521 (4) + BE/ME, Eq vol, Firm T 1.5843 0.2081 (6) IG 0.8467 0.1442 0.4670 0.0412 (5) + Liquidity 6.1588 0.0131 (7) Junk 3.1838 0.4155 0.9848 0.1280 T, Rating, time FE, firm FE 20.3059 0.0000 (8) Junk 3.1345 0.4391 1.1060 0.1481 (7) + BE/ME, Eq vol, Firm T 14.6705 0.0001 (9) Junk 3.4480 0.4493 0.8510 0.1292 (8) + Liquidity 25.1677 0.0000 B. Hedge ratios Model Empirical Model test Avg SE Coeff SE Controls (interacted with |$r_E$|⁠) |$\chi^2$| p (1) Full 0.0820 0.0082 0.0732 0.0154 T, Rating, firm FE 0.2601 0.6100 (2) Full 0.0800 0.0081 0.0852 0.0185 (1) + BE/ME, Eq vol, Firm T 0.0660 0.7972 (3) Full 0.0831 0.0082 0.0704 0.0159 (2) + Liquidity 0.5147 0.4731 (4) IG 0.0659 0.0079 0.0735 0.0171 T, Rating, firm FE 0.1792 0.6720 (5) IG 0.0658 0.0080 0.0856 0.0202 (4) + BE/ME, Eq vol, Firm T 0.8850 0.3468 (6) IG 0.0674 0.0080 0.0665 0.0213 (5) + Liquidity 0.0019 0.9657 (7) Junk 0.1886 0.0150 0.0777 0.0179 T, Rating, firm FE 21.1105 0.0000 (8) Junk 0.1818 0.0137 0.0881 0.0218 (7) + BE/ME, Eq vol, Firm T 12.0823 0.0005 (9) Junk 0.1924 0.0137 0.0832 0.0130 (8) + Liquidity 33.3422 0.0000 A. Yield spreads . . . Model . Empirical . . Model test . . . Avg . SE . Coeff . SE . Controls . |$\chi^2$| . p . (1) Full 1.1253 0.1678 0.7001 0.0567 T, Rating, time FE, firm FE 5.5384 0.0186 (2) Full 1.0993 0.1679 0.6625 0.0576 (1) + BE/ME, Eq vol, Firm T 5.5009 0.0190 (3) Full 1.1811 0.1733 0.4773 0.0504 (2) + Liquidity 13.4658 0.0002 (4) IG 0.8110 0.1453 0.6448 0.0537 T, Rating, time FE, firm FE 1.3180 0.2509 (5) IG 0.8110 0.1458 0.6229 0.0521 (4) + BE/ME, Eq vol, Firm T 1.5843 0.2081 (6) IG 0.8467 0.1442 0.4670 0.0412 (5) + Liquidity 6.1588 0.0131 (7) Junk 3.1838 0.4155 0.9848 0.1280 T, Rating, time FE, firm FE 20.3059 0.0000 (8) Junk 3.1345 0.4391 1.1060 0.1481 (7) + BE/ME, Eq vol, Firm T 14.6705 0.0001 (9) Junk 3.4480 0.4493 0.8510 0.1292 (8) + Liquidity 25.1677 0.0000 B. Hedge ratios Model Empirical Model test Avg SE Coeff SE Controls (interacted with |$r_E$|⁠) |$\chi^2$| p (1) Full 0.0820 0.0082 0.0732 0.0154 T, Rating, firm FE 0.2601 0.6100 (2) Full 0.0800 0.0081 0.0852 0.0185 (1) + BE/ME, Eq vol, Firm T 0.0660 0.7972 (3) Full 0.0831 0.0082 0.0704 0.0159 (2) + Liquidity 0.5147 0.4731 (4) IG 0.0659 0.0079 0.0735 0.0171 T, Rating, firm FE 0.1792 0.6720 (5) IG 0.0658 0.0080 0.0856 0.0202 (4) + BE/ME, Eq vol, Firm T 0.8850 0.3468 (6) IG 0.0674 0.0080 0.0665 0.0213 (5) + Liquidity 0.0019 0.9657 (7) Junk 0.1886 0.0150 0.0777 0.0179 T, Rating, firm FE 21.1105 0.0000 (8) Junk 0.1818 0.0137 0.0881 0.0218 (7) + BE/ME, Eq vol, Firm T 12.0823 0.0005 (9) Junk 0.1924 0.0137 0.0832 0.0130 (8) + Liquidity 33.3422 0.0000 This table compares extended Merton model-implied yield spreads and hedge ratio effects with empirically estimated effects. Panel A considers yield spreads and panel B hedge ratios. The model columns display the average effect after calculating observation-by-observation model effects and the standard error is for the average. The empirical columns have estimates that correspond to the coefficient on proportion due prior in the yield spread regressions and |$r_E\times\text{proportion due prior}$| in the hedge ratio regressions. For the hedge ratio regressions, the controls include both interaction terms with |$r_E$| and also the controls on their own. The last two columns present the results of Wald test for which the null hypothesis is that the model and empirical effects are equal. Standard errors are two-way clustered by firm and time. Open in new tab Similarly, we test model hedge ratios in panel B of Table 9. These tests also show that the model fits better for investment-grade bonds than for junk bonds. In all of the specifications, we see that the model cannot be rejected for investment-grade bonds, whereas the model is strongly rejected for junk bonds. In contrast to the yield spread results, we find that the extended Merton model cannot be rejected for the full sample. The strong rejection of the extended Merton model for junk bonds as compared to its reasonable fit for investment-grade bonds is consistent with complexities in junk bond debt structures that the model is failing to capture. We now turn to discussing some of these complexities, including covenants, firms extremely close to bankruptcy, and risk shifting. In the extended Merton model, we abstract away from modeling covenants. Covenants on public debt are relatively weak and often are only incurrence covenants (Chava and Roberts 2008 and Maxwell and Shenkman 2010). Thus, firms can relatively easily avoid violating such covenants in most cases. In our sample, the most common types of covenants are (1) consolidation/merger, (2) negative pledge, and (3) asset sales covenants. Consolidation/merger covenants restrict mergers and should have little direct effect on de facto seniority. Negative pledge covenants prevent secured debt issuance, showing that bondholders are concerned about explicit seniority. Asset sales covenants restrict the use of proceeds when assets are sold (Maxwell and Shenkman 2010). Typically, proceeds can be used to permanently repay debt or invest in replacement assets. Thus, such covenants explicitly allow firms to sell assets to repay debt, consistent with the extended Merton model where maturing debt is paid off from firm value.32 Although bond covenants are unlikely to undo de facto seniority, Rauh and Sufi (2010) show that junk firms are likely to have heterogeneous debt and, in particular, bank debt with tight covenants along with nonbank debt with loose covenants. In contrast, high credit quality firms have little bank debt. More importantly, bank debt often has financial covenants which firms can violate through poor accounting performance even if they are still able to service debt. Common examples include maximum debt-to-EBITDA, minimum interest coverage, and minimum fixed charge coverage ratios. This means that there is likely to be an earlier default trigger. Hence, the combination of the fact that junk firms are much more likely to have bank debt along with the earlier contingent control that bank debt gives to creditors suggests that the empirical effect of de facto seniority is likely to be muted relative to what the extended Merton model suggests. A second reason for why the empirical effect of de facto seniority for junk firms may be muted relative to the model is what happens when a firm is very close to bankruptcy. The model suggests that as credit quality worsens, the effect of de facto seniority becomes stronger. This is true for most levels of credit quality, but once a firm crosses the threshold of being very likely to default in the near future, almost all of the value of its bonds comes from expected recovery. Since recovery in bankruptcy is based on explicit seniority and Guha (2002) shows that explicit priority is usually followed in Chapter 11, de facto seniority no longer applies. In the extended Merton model, this corresponds to cases where |$K_1$|⁠, the face value of the first bond, is high enough that the firm is likely to default the first time it has to service a bond. Reality is more complicated as a combination of a shortage of cash flows with an inability to liquidate assets to service debt could lead to default even if a firm’s balance sheet suggests that its assets are large enough to service debt. Using the subsample of firm-months where firms have a distance-to-default of less than 1.5 (roughly the 1% of observations closest to default), we find that the relation between yield spreads and proportion due prior is statistically and economically insignificant.33 Hence, the de facto seniority effect does disappear for the firms most likely to default. To the extent that the model cannot fully and precisely account for the firms that are at the extreme for default probability, the model is likely to significantly overestimate the de facto seniority effect in such cases. Finally, the model is less likely to fit hedge ratios for junk firms because managers at junk firms may choose to risk shift, decoupling corporate bond returns from equity returns. When managers take actions that affect firm value or the market reassesses a company’s expected cash flows, bondholders and equityholders are affected in the same direction. In the extended Merton model, exposure to firm value shocks are what drives the comovement of corporate bonds and equities. Asset volatility, strictly speaking, is a constant in the model. In reality, managers may take risk-shifting actions. Around the period when such an action is taken, equity returns are likely to be positive and bond returns negative. The hedge ratio, estimated over many periods, is then lower compared to what a constant volatility model would suggest. Risk shifting is much more likely to occur in distressed firms (Eisdorfer 2008), potentially helping to explain why the proportion due prior effect is only slightly higher for junk firms than investment-grade firms when we examine hedge ratios in Table 8, even though the model suggests that hedge ratios should be much higher for junk firms. Overall, our results suggest that the extended Merton model is a better representation of investment-grade firms than junk firms. Junk firms have more complex capital structures, stricter covenants, extreme observations, and managerial decisions that make our simple extension of the Merton model deviate further from reality than for investment-grade firms. Although the extended Merton model produces de facto seniority effects roughly in line with empirical estimates for investment-grade firms, the model is, of course, not a full representation of all of the complexities of a firm’s debt structure. Similar to the results on hedge ratios by Schaefer and Strebulaev (2008), our results show that a relatively simple model can give a first order approximation of an empirical effect. 4. Conclusion In this paper, we have studied the effect of a bond’s place in its issuer’s maturity structure on credit risk. Using an extension of the Merton model, we illustrated the mechanism by which a bond that is late in its issuer’s maturity structure may have more credit risk than a bond due early in its issuer’s maturity structure. The intuition follows from the sequential repayment of debt. Even if a firm is in some financial trouble, it may be able to pay back the first few issues of debt that mature. However, it may be unable to pay back later issues as its credit quality further deteriorates. In reality, the repayment of earlier debt may even put the firm in a more precarious position. The de facto seniority concept has implications for both the levels of bond prices and returns. Thus, we studied the effects of a bond’s place in its issuer’s maturity structure on both yield spreads and hedge ratios. Our main empirical finding is that bonds due later in their issuer’s maturity structure have both higher yield spreads and higher hedge ratios. This result is robust to including a series of controls for credit risk, illiquidity, the bond’s maturity, and the average maturity of all of the issuer’s bonds. Benchmarking the model-implied de facto seniority effect against our empirical estimates, we found that the extended Merton model is rejected for junk bonds, but is not rejected for investment-grade bonds in most specifications. These results are consistent with the literature showing that the capital structure of junk firms is significantly more complicated than for investment-grade firms. The extended Merton model, largely designed to provide intuition and motivate the de facto seniority effect, deviates more from the complex structure of junk-rated firms. Overall, our results suggest a strong de facto seniority effect in the corporate bond market reflected in both the levels of bond prices and the dynamics of returns. Appendix A. Extending the Merton Model In our extension of the Merton model, the firm has three zero-coupon bond issues with face values |$K_1$|⁠, |$K_2$|⁠, and |$K_3$|⁠, and maturities |$T_1 < T_2 < T_3$|⁠. All of the bonds share the same seniority. For example, suppose that the firm defaults prior to |$T_1$|⁠. The proportion of the total recovered value of the firm paid to bond 1 is |$\frac{K_1}{K_1 + K_2 + K_3}$|⁠. In addition, we also allow for a proportion |$L$| of the residual firm value to be lost in bankruptcy. That is, if the value of the firm at default is |$V_{default}$|⁠, the proportion payable to bondholders is |$(1-L)V_{default}$|⁠. A.1 Equity Value \begin{align*} E = E_0^Q\left[e^{-rT_3}{\rm 1}\kern-0.24em{\rm I}_{V_1 > K_1}{\rm 1}\kern-0.24em{\rm I}_{V_2>K_2}{\rm 1}\kern-0.24em{\rm I}_{V_3>K_3}\left(V_3 - K_3\right)\right] \end{align*} The firm value process is lognormal \begin{align} &V_1 = V_0 \exp\left[\left(r-\frac{1}{2}\sigma_v^2\right)T_1 + \sigma_v\sqrt{T_1} w_1\right],\\ \notag & \text{where} w_1 \sim N(0,1). \end{align}(A.1) Similarly, \begin{align*} &V_2 = (V_1-K_1) \exp\left[\left(r-\frac{1}{2}\sigma_v^2\right)(T_2-T_1) + \sigma_v\sqrt{T_2 - T_1} w_2\right]\\\notag & \text{if} V_1 > K_1\\ \notag & \text {and}\\ \notag &V_3 = (V_2-K_2) \exp\left[\left(r-\frac{1}{2}\sigma_v^2\right)(T_3-T_2) + \sigma_v\sqrt{T_3 - T_2} w_3\right]\\\notag & \text{if} V_2 > K_2\text{.} \end{align*} In particular, note that if the firm is solvent at |$T_i$|⁠, then |$K_i$| is paid out of firm value for the maturing bond. After some algebra, it can be shown that \begin{align} E &= \frac{e^{-rT_2}}{2\pi}\int_{-d_2}^{\infty} \int_{-\tilde{d}_2}^{\infty} \exp\left(-\frac{1}{2}z_2^2 - \frac{1}{2}z_1^2\right)(V_2 - K_2)N\left(\tilde{\tilde{d}}_1\right) dz_2 dz_1\\\notag &-K_3\frac{-e^{-rT_3}}{2\pi}\int_{-d_2}^{\infty} \int_{-\tilde{d}_2}^{\infty}\exp\left(-\frac{1}{2}z_2^2 - \frac{1}{2}z_1^2\right)N\left(\tilde{\tilde{d}}_2\right)dz_2 dz_1,\\ \notag & \text{where} \tilde{\tilde{d}}_2 = \frac{\ln\left(\frac{V_2-K_2}{K_3}\right)+\left(r-\frac{1}{2}\sigma_v^2\right)(T_3-T_2)}{\sigma_v\sqrt{T_3 - T_2}}, \tilde{\tilde{d}}_1 = \tilde{\tilde{d}}_2 + \sigma_v\sqrt{T_3 - T_2}\\\notag &\text{and }\tilde{d}_2 = \frac{\ln\left(\frac{V_1-K_1}{K_2}\right)+\left(r-\frac{1}{2}\sigma_v^2\right)(T_2-T_1)}{\sigma_v\sqrt{T_2 - T_1}}. \end{align}(A.2) A.2 Bond Value Our focus is on the intermediate bond with maturity |$T_2$| and face value |$K_2$|⁠. The price of this bond can be written as the sum of three expectations, which correspond to (1) solvency at |$T_2$|⁠, (2) default at |$T_1$|⁠, and (3) default at |$T_2$|⁠. The value of each of these pieces corresponds to |$E_0^Q\left[e^{-rT_2}{\rm 1}\kern-0.24em{\rm I}_{V_1 > K_1}{\rm 1}\kern-0.24em{\rm I}_{V_2>K_2}K_2\right]$|⁠; |$E_0^Q\left[e^{-rT_1}{\rm 1}\kern-0.24em{\rm I}_{V_1 < K_1}(1-L)V_1\frac{K_2}{K_1+K_2+K_3}\right]$|⁠; and |$E_0^Q\left[e^{-rT_2}{\rm 1}\kern-0.24em{\rm I}_{V_1 > K_1}{\rm 1}\kern-0.24em{\rm I}_{V_2<K_2}(1-L)V_2\frac{K_2}{K_2 + K_3}\right]$|⁠. After some algebra, it can be shown that the three pieces are |$K_2\frac{e^{-rT_2}}{\sqrt{2\pi}}{\int}_{-d_2}^{\infty}\exp\left(-\frac{1}{2}z_1^2\right) N\left(\tilde{d}_2\right) dz_1$|⁠; |$V_0(1-L)\frac{K_2}{K_1+K_2+K_3}N(-d_1)$|⁠; and |$\frac{K_2(1-L)e^{-rT_1}}{(K_2+K_3)\sqrt{2\pi}}{\int}_{-d_2}^{\infty} \exp\left(-\frac{1}{2}z_1^2\right)\left(V_1 - K_1\right)N\left(-\tilde{d}_1\right)dz_1$|⁠. Corporate bond yields can then be calculated from prices and yield spreads are attained from subtracting off the relevant Constant Maturity Treasury yield. A.3 Hedge Ratios In principle, numerical partial derivatives can be calculated once we have bond prices. However, in our implementation of the extended Merton model, we find that numerical partial derivatives calculated for prices that are determined though numerical integration can become quite imprecise. Thus, we apply Leibniz’s rule to first calculate partial derivatives as functions of integrals (much like the bond prices themselves) before numerically integrating. In particular, the three pieces of the partial derivative of the bond price (from A.2) with respect to firm value (⁠|$\partial B/\partial V$|⁠) are |$K_2 e^{-rT_2}{\int}_{-d_2}^{\infty}\frac{1}{\sqrt{2\pi}}exp\left(-\frac{1}{2}z_1^2\right)n\left(\tilde{d}_2\right)\frac{V_1}{V_0}\frac{1}{(V_1-K_1)\sigma_v\sqrt{T_2-T_1}}dz_1$|⁠; |$\frac{(1-L)K_2N(-d_1)}{K_1+K_2+K_3}-\frac{(1-L)K_2n(-d_1)}{(K_1+K_2+K_3)\sigma_v\sqrt{T_1}}$|⁠; and |$\frac{(1-L)K_2e^{-rT_1}}{K_2+K_3}\Big[{\int}_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}}exp\left(-\frac{1}{2}z_1^2\right)\frac{V_1}{V_0}N \left(-\tilde{d}_1\right) dz_1$||$-{\int}_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}}exp\left(-\frac{1}{2}z_1^2\right)\frac{(V_1-K_1)n(-\tilde{d}_1)V_1}{\sigma_v\sqrt{T_2-T_1}(V_1 - K_1)V_0}dz_1\Big]$|⁠. The hedge ratio can then be calculated by taking |$\frac{\partial B}{\partial V}\frac{V}{B}$| and dividing by the numerically calculated |$\frac{\partial E}{\partial V}\frac{V}{E}$|⁠. B. Other Models B.1 Geske Model The extended Merton model in Section 1 and Appendix A is based on the assumption that upon bond maturity, the face value of the bond is paid out of the value of the firm. There is no explicit choice for equityholders to make (though given the choice, they would always choose to continue the firm as the value of their residual claim if the firm defaults is zero). Another class of structural models of default, endogenous default models, force equityholders to directly bear the cost of payouts (coupons, face value payments, or both) to bondholders.34 Thus, at each moment that there is a payoff, equityholders make a decision by trading-off the value they must pay to continue the firm against their claim to the value of the firm as a going concern. This pay-in induces equityholders to consider the full maturity structure of the firm when deciding whether to continue the firm. Here, we consider the Geske (1977) model.35 In the Geske model, like in most structural models of default, equityholders are residual claimants of the firm. That is, if the firm survives to the very last promised payment to bondholders at time |$T$|⁠, equityholders receive |$\max(0,V_T - K_T)$|⁠, where |$V_T$| is the value of the firm at |$T$| and |$K_T$| is the last promised payment. At any prior time |$t$| with a payment to bondholders of |$K_t$|⁠, equityholders must decide whether to pay |$K_t$| and allow the firm to continue or to force the firm to default and put the firm to bondholders. Effectively, equityholders are making an optimal decision based on the market value of |$\max(0,V_T - K_T)$| at |$t$| compared to the payment |$K_t$|⁠. Similar to our extension of the Merton model, we include three zero-coupon bonds with maturities |$T_1 < T_2 < T_3$| and focus on the intermediate maturity bond, while varying the proportion of the firm’s debt value due to the first and third bonds. We again consider two scenarios, the first where the bond maturing at |$T_3$| accounts for 80% of the firm’s debt value and the second where the bond maturing at |$T_1$| accounts for 80% of the firm’s debt value. This allows us to compare the case in which most of the firm’s debt is due after the intermediate maturity bond and the case in which most of the firm’s debt is due before the intermediate maturity bond. Rather than providing the full set of Geske pricing formulas for general cases, we instead focus on the intuition of the Geske model under the special case that we implement, which is best understood through a recursive framework. In the Geske model, firm value follows a geometric Brownian motion like in the Merton model. Also, like in the Merton model, if the firm survives until there is only one bond outstanding (the bond with maturity |$T_3$| in our example), equity is a residual claim and receives a payoff of |$\max(0,V_3 - K_3)$| at |$T_3$|⁠. Consider the value of equity and the long maturity bond at |$T_2$| and suppose that equityholders have just decided to pay the face value of the intermediate maturity bond, |$K_2$|⁠, to continue the firm. This becomes the standard Merton model as equityholders have no remaining decisions to make. The value of equity is now \begin{align}\label{equity_geske} E_2 &= V_2 N(d_1) - K_3e^{-r(T_3-T_2)}N(d_2)\text{,}\\\notag d_1 &= \frac{\ln\left(\frac{V_2}{K_3}\right)+\left(r+\frac{1}{2}\sigma_v^2\right)\left(T_3-T_2\right)}{\sigma_v\sqrt{T_3-T_2}}, d_2 = d_1 - \sigma_v\sqrt{T_3-T_2}. \end{align}(A.3) That is, equity is a call option on the firm and the long maturity bond is a risk-free bond short a put option on the firm. We can now consider the optimal decision of equityholders at |$T_2$|⁠. The decision that equityholders make is simple. If the value of equity given in Equation (A.3) is greater than |$K_2$|⁠, they pay |$K_2$| and continue the firm. Otherwise, equityholders force the firm to go bankrupt. Even if |$K_2$| is small, equityholders may still choose bankruptcy because |$K_3$| is very large and the continuation value of equity is very small. Equityholders’ decision at |$T_1$| is similar, though the trade-off that equityholders make is between the payment of the face value of the short maturity bond, |$K_1$|⁠, and the value of equity as a going concern which is a compound call option. Because equityholders take into account the full maturity structure of the firm’s debt, even bonds that mature relatively early can have significant exposure to underlying firm value. Intuitively, even if the debt due early has a low face value, equityholders know that they will only receive a nonzero payoff in the future if the firm value is high enough to cover all face value payments in the future. Having large face values due in the future would be equivalent to buying an out-of-the-money call option, so even a relatively small pay-in may induce default. As noted by Eom, Helwege, and Huang (2004), the Geske formula is not straightforward to implement accurately, particularly for long maturity bonds. Thus, we follow Huang (1997) and Eom, Helwege, and Huang (2004) and implement the Geske model using a Cox, Ross, and Rubinstein (1979) binomial tree, and the equity and bond values (from which we calculate yields), as well as the hedge ratio, are determined numerically. We also verify the accuracy of this numerical implementation by running simulations. In Figure A.1, we present the yield spreads and hedge ratios for the intermediate maturity bond for the scenarios where (1) the bond is early in its issuer’s maturity structure and (2) the bond is late in its issuer’s maturity structure. Recall that for the extended Merton model, the yield spread and hedge ratio are generally lower in the first scenario. The intuition follows from the fact that if most of the debt is due after |$T_2$|⁠, the firm would very likely have enough value to pay bondholders at |$T_2$| directly from firm value. Thus, the intermediate bond has a higher price and lower sensitivity to firm value. In the Geske model, the intuition is less clear. Even if most of the firm’s debt is due at |$T_3$|⁠, equityholders may still choose to default at |$T_2$| as their payoff at |$T_3$| contingent on continuation, |$\max(0,V_3 - K_3)$|⁠, may have a low expected value due to the large amount of debt due at |$T_3$|⁠. Intuitively, this is akin to an investor choosing whether to pay a price for an out-of-the-money call option. The out-of-the-money call option will be cheaper than an in-the-money call, but the investor is not necessarily more likely to buy it. We find that for the same set of parameters as the extended Merton model in Section 1, the Geske model generates slightly lower yield spreads and hedge ratios for scenario 2 than scenario 1 for almost all leverage-asset volatility combinations.36 Effectively, the significant discrete pay-in induces equityholders to consider the firm’s full maturity structure when making continuation decisions. Figure A.1 Open in new tabDownload slide Yield spreads and hedge ratios for the Geske (1977) model. Yield spreads are reported in basis points and hedge ratios in percentage. The cyan surfaces represent the case in which a bond is late in its issuer’s maturity structure and the red surfaces with point markers represent the case in which a bond is early in its issuer’s maturity structure. B.2 Extending the Geske Model to Incorporate Partial Pay-In Thus far, we have considered an extended Merton model in Section 1 and Appendix A and the Geske model in Appendix B.1. These two models make very different assumptions about how maturing bonds are paid off. In the extended Merton model, a maturing bond is paid from firm assets, whereas in the Geske model, equityholders pay the full face value of the maturing bond. Here, we consider an intermediate case, allowing the maturing bond to be paid partially from firm assets and partially by equityholders. Define |$\alpha$| to be the proportion of the maturing bond face value that can be paid from firm assets. If |$\alpha K_t \leq V_t$|⁠, |$\alpha K_t$| is paid from firm assets and equityholders decide whether to pay |$(1-\alpha)K_t$| to continue the firm. The extended Merton and Geske models are the two special cases that correspond to |$\alpha = 1$| and |$\alpha = 0$|⁠, respectively. In the extended Merton model, equityholders always continue the firm if |$V_t > K_t$|⁠, whereas in the Geske model, equityholders sometimes choose to default even if |$V_t > K_t$| because the pay-in to continue the firm is too high relative to the value of their potential future payoff. Whether decreasing the equity pay-in significantly increases the likelihood of equityholders continuing the firm is unclear as the decreased pay-in comes at the cost of decreasing firm value. To illustrate the point, suppose that a firm has a zero-coupon bond maturing at time |$T_2$| with a face value of 20 and another zero-coupon bond maturing at time |$T_3 = T_2+3$| with a face value of 30. Suppose that the firm value is 45 at |$T_2$| and it only has these two bond issues outstanding. Naturally, at |$\alpha = 1$|⁠, equityholders always continue the firm. If |$\alpha = 0$|⁠, equityholders need to decide whether to pay 20 in exchange for a call option maturing in three years with a strike of 30 and a current underlying asset value of 45. In this case, the value of the call option is 17.25 and equityholders choose to let the firm default. Consider instead |$\alpha = 0.5$|⁠. In this case, equityholders only need to pay |$(1-\alpha)\times20=10$| to continue the firm. However, the other 10 is paid out of firm value, so equityholders are deciding whether to buy a call option with a strike of 30 and an underlying asset value of 35. Despite the decreased pay-in, equityholders still choose to let the firm default because the value of the call option is only worth 8.52 because the option is less in-the-money than before. In Figure A.2, we plot the value of the call option and the cost to equityholders to continue the firm. By definition, between the two extreme cases of |$\alpha = 0$| and |$\alpha = 1$|⁠, there is an |$\alpha \in [0,1]$| where equityholders are indifferent. In this example, this |$\alpha$| is slightly less than 0.8. Figure A.2 Open in new tabDownload slide Dashed line is the amount that equityholders need to pay-in to continue the firm for different levels of |$\alpha$| in the extended Geske model. Solid curve is the value of the call option that equityholders would hold if they choose to continue the firm. In Figures A.3 and A.4, we plot the difference between the yield spreads and hedge ratios of the intermediate maturity bond when most of the firm’s debt is due before the bond and when most of the firm’s debt is due after the bond for different levels of |$\alpha$|⁠, using the same parameters from Section 1. For low values of |$\alpha$|⁠, the difference in yield spreads and hedge ratios between the two scenarios is close to 0. At |$\alpha = 0.8$|⁠, we begin to see some evidence of de facto seniority, as the yield spread and hedge ratio of the intermediate maturity bond when most of the firm’s debt is due before it becomes higher. What drives the relatively high |$\alpha$| needed to generate de facto seniority is that there is typically a wide set of asset values where |$V_i$| is not much larger than |$K_i$| and the continuation value of equity is low. Significant required pay-in by equityholders would result in equityholders choosing default, whereas a zero pay-in (⁠|$\alpha = 1$|⁠) would lead to continuation. Figure A.3 Open in new tabDownload slide Yield spreads for the the extended Geske model. The panels plot differences in yield spreads between bonds due late in a firm’s maturity structure and bonds due early in a firm’s maturity structure. |$\alpha$| represents the proportion of maturing debt that is paid by liquidating firm assets. Figure A.4 Open in new tabDownload slide Hedge Ratios for the the extended Geske model. The panels plot differences in hedge ratios between bonds due late in a firm’s maturity structure and bonds due early in a firm’s maturity structure. |$\alpha$| represents the proportion of maturing debt that is paid by liquidating firm assets. B.3 Leland-Toft Model As a final alternative model, we consider a Leland and Toft (1996) model where equityholders continuously rollover maturing debt. All debt is rolled over at a prespecified (exogenously given) maturity. An exogenously specified fraction |$\delta$| of firm value is paid out to debtholders and equityholders. If the sum of proceeds from newly issued debt and |$\delta V$| is not large enough to cover coupons to debtholders and the face value of maturing debt, equityholders may choose to inject capital to continue the firm. The decision about whether or not to inject capital to continue leads to an endogenously determined default boundary. Compared to the previously explored models in this paper, the Leland-Toft model adds a number of layers of complexity. First, debt is rolled over rather than allowed to mature like in the extended Merton, Geske, and extended Geske models explored earlier. Second, although default is endogenous like in the Geske model, injections of capital do not always occur. In the Geske model, shareholders always pay to retire existing debt. In the Leland-Toft model, shareholder payments only occur if there is a cash shortfall. Third, shareholder injections of cash (when necessary) tend to be small and continuous in the Leland-Toft model, rather than large and discrete like in the Geske model. Fourth, the default boundary is endogenously determined in the Leland-Toft model, in contrast to all of the models previously examined in this paper. While such dynamics are interesting and potentially more realistic, they also complicate the intuition due to the large number of simultaneously moving parts. To implement the Leland and Toft (1996) model, we follow equation (13) in their paper to determine the optimal default boundary. Bond pricing follows from equation (3) of their paper and we also choose the coupon rate so that newly issued debt sells at par (equation (14) in their paper). We choose |$r = 0.03$|⁠, a firm-level payout ratio |$\delta = 0.05$|⁠, |$\alpha=0.5$| (the deadweight loss of bankruptcy) and |$\tau=0.35$| (tax rate). We try a number of asset volatility (⁠|$\sigma_v$|⁠) and book leverage combinations. Our analysis focuses on a comparison of a 5-year bond in two examples. First, we consider a firm where all newly issued debt has a 5-year maturity. In particular, 20% of the firm’s debt is rolled over each year. Next, we consider a firm where all newly issued debt has a 15-year maturity. In the former case, the 5-year bond is due late in its firm’s maturity structure (i.e., is de facto junior) and in the latter, it is due relatively early. In Figure A.5, we plot the yield spreads and hedge ratios of the two scenarios. We find that yield spreads are higher when a bond is de facto junior. Our results on hedge ratios show that for asset volatilities and leverages at levels consistent with typical firms in our sample (asset volatility around 0.2 and leverage around 0.4), bonds that are due relatively late in their firm’s maturity structure have higher hedge ratios. The intuition behind these results is somewhat more complicated than the previous models due to the additional moving parts. Like the extended Merton model, equityholders will tend to continue the firm when possible, making the bonds due early in the maturity structure safer than bonds due later in the maturity structure. However, there is occasionally the need for equityholders to pay-in, making equityholders more likely to default than in the extended Merton model. The intuition is similar to the extended Geske model. Pay-ins are less frequent than the Geske model, but more frequent than the extended Merton model (which has no equity pay-in), making the gap in yield spreads an intermediate case between the Geske and extended Merton cases. Figure A.5 Open in new tabDownload slide Yield spreads and hedge ratios for the Leland and Toft (1996) model. Yield spreads are reported in basis points and hedge ratios in percentage. The cyan surfaces with circular markers represent cases where a bond is due late in its issuer’s maturity structure and the white surfaces with point markers represent cases where a bond is due early in its issuer’s maturity structure. The intuition for hedge ratios in the Leland-Toft model is significantly more complicated. A bond that is due relatively early in its firm’s maturity structure is less sensitive to asset value because it is safer. In particularly, its hedge ratio relative to firm assets, |$\frac{\partial \ln B}{\partial \ln V}$| is lower if it is early in its firm’s maturity structure. However, the sensitivity of equity to firm value also decreases when we move from a firm rolling over 5-year bonds to a firm rolling over 15-year bonds. This arises from the fact that the firm rolling over 15-year bonds chooses a lower default boundary than a firm rolling over 5-year bonds. This causes |$\frac{\partial \ln E}{\partial \ln V}$| to be significantly lower in some cases with higher leverage and for |$\frac{\partial \ln B}{\partial \ln E}$| to be higher for the 5-year bond issued by a firm rolling over 15-year bonds than the 5-year bond issued by a firm rolling over 5-year bonds. The general prediction of the Leland-Toft model is that a bond that is later in its issuer’s maturity structure will tend to have higher yield spreads and higher hedge ratios, though the effect is not as uniform as the extended Merton model predicts. C. Alternative Controls In Sections 3.1 and 3.2, we present empirical results showing that yield spreads and hedge ratios are higher for bonds that are de facto junior. Here, we consider the robustness of our results to alternative control variables. Our primary empirical analysis uses equity volatility and BE/ME as two simple credit controls that are meant to aggregate market information about credit risk. In the results that follow, we replace equity volatility and BE/ME with |$\frac{K}{V}$| and asset volatility (⁠|$\sigma_v$|⁠), two credit variables that are directly motivated by structural models of default. To calculate |$\frac{K}{V}$| and asset volatility, we follow Campbell, Hilscher, and Szilagyi (2008) and calculate them from two equations, \begin{align}\label{eq:call_option} E = V N(d_1) - K e^{-rT} N(d_2) \end{align}(A.4) and \begin{align}\label{eq:mer_eq_vol} \sigma_E = N(d_1)\frac{V}{E}\sigma_v. \end{align}(A.5) Furthermore, we also augment our empirical analysis to allow for a number of nonlinear controls. In addition to controlling for both firm and time fixed effects like in panel B of Table 5, we include ratings fixed effects (rather than numerically coded ratings). We also allow for squared terms for time-to-maturity (T), |$\frac{K}{V}$|⁠, and asset volatility. In Table A.1, we find that the effect of de facto seniority on yield spreads is largely similar to Table 5. Table A.1 Yield spreads with nonlinear controls . Full . IG . Junk . Proportion due prior 0.46 0.39 0.77 [8.16] [9.04] [5.41] T 0.0113 0.0137 0.0111 [4.56] [5.64] [1.30] T|$^2$| –0.0001 –0.0001 –0.0001 [–2.31] [–3.03] [–0.71] K/V 1.87 2.22 2.37 [3.07] [3.69] [1.90] (K/V)|$^2$| 3.48 2.28 2.98 [3.35] [1.76] [2.13] Asset vol –0.0020 0.0046 0.0033 [–0.33] [0.78] [0.18] Asset vol|$^2$| 0.0003 0.0002 0.0002 [3.99] [2.40] [0.75] Firm T –0.01 –0.01 0.03 [–1.86] [–1.07] [1.90] Amihud 0.03 0.03 0.03 [3.13] [5.11] [3.02] IRC 0.50 0.46 0.37 [8.55] [7.44] [4.26] Amihud vol 0.02 0.02 0.05 [6.31] [5.57] [5.65] IRC vol –0.02 –0.04 0.17 [–0.70] [–1.52] [2.66] |$R^2$| 0.83 0.78 0.81 Observations 205,822 177,311 28,511 . Full . IG . Junk . Proportion due prior 0.46 0.39 0.77 [8.16] [9.04] [5.41] T 0.0113 0.0137 0.0111 [4.56] [5.64] [1.30] T|$^2$| –0.0001 –0.0001 –0.0001 [–2.31] [–3.03] [–0.71] K/V 1.87 2.22 2.37 [3.07] [3.69] [1.90] (K/V)|$^2$| 3.48 2.28 2.98 [3.35] [1.76] [2.13] Asset vol –0.0020 0.0046 0.0033 [–0.33] [0.78] [0.18] Asset vol|$^2$| 0.0003 0.0002 0.0002 [3.99] [2.40] [0.75] Firm T –0.01 –0.01 0.03 [–1.86] [–1.07] [1.90] Amihud 0.03 0.03 0.03 [3.13] [5.11] [3.02] IRC 0.50 0.46 0.37 [8.55] [7.44] [4.26] Amihud vol 0.02 0.02 0.05 [6.31] [5.57] [5.65] IRC vol –0.02 –0.04 0.17 [–0.70] [–1.52] [2.66] |$R^2$| 0.83 0.78 0.81 Observations 205,822 177,311 28,511 The dependent variable in this table is yield spread in percentage. All regressions include three sets of fixed effects: time fixed effects, firm fixed effects, and ratings fixed effects (by notch). Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. K/V (in decimals) and Asset vol (in percentage) are calculated following Campbell, Hilscher, and Szilagyi (2008). Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. Amihud, Amihud vol, IRC, and IRC vol are the Amihud measure, the volatility of the Amihud measure, implied round-trip cost, and the volatility of implied round-trip cost, respectively. All four are calculated following Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Table A.1 Yield spreads with nonlinear controls . Full . IG . Junk . Proportion due prior 0.46 0.39 0.77 [8.16] [9.04] [5.41] T 0.0113 0.0137 0.0111 [4.56] [5.64] [1.30] T|$^2$| –0.0001 –0.0001 –0.0001 [–2.31] [–3.03] [–0.71] K/V 1.87 2.22 2.37 [3.07] [3.69] [1.90] (K/V)|$^2$| 3.48 2.28 2.98 [3.35] [1.76] [2.13] Asset vol –0.0020 0.0046 0.0033 [–0.33] [0.78] [0.18] Asset vol|$^2$| 0.0003 0.0002 0.0002 [3.99] [2.40] [0.75] Firm T –0.01 –0.01 0.03 [–1.86] [–1.07] [1.90] Amihud 0.03 0.03 0.03 [3.13] [5.11] [3.02] IRC 0.50 0.46 0.37 [8.55] [7.44] [4.26] Amihud vol 0.02 0.02 0.05 [6.31] [5.57] [5.65] IRC vol –0.02 –0.04 0.17 [–0.70] [–1.52] [2.66] |$R^2$| 0.83 0.78 0.81 Observations 205,822 177,311 28,511 . Full . IG . Junk . Proportion due prior 0.46 0.39 0.77 [8.16] [9.04] [5.41] T 0.0113 0.0137 0.0111 [4.56] [5.64] [1.30] T|$^2$| –0.0001 –0.0001 –0.0001 [–2.31] [–3.03] [–0.71] K/V 1.87 2.22 2.37 [3.07] [3.69] [1.90] (K/V)|$^2$| 3.48 2.28 2.98 [3.35] [1.76] [2.13] Asset vol –0.0020 0.0046 0.0033 [–0.33] [0.78] [0.18] Asset vol|$^2$| 0.0003 0.0002 0.0002 [3.99] [2.40] [0.75] Firm T –0.01 –0.01 0.03 [–1.86] [–1.07] [1.90] Amihud 0.03 0.03 0.03 [3.13] [5.11] [3.02] IRC 0.50 0.46 0.37 [8.55] [7.44] [4.26] Amihud vol 0.02 0.02 0.05 [6.31] [5.57] [5.65] IRC vol –0.02 –0.04 0.17 [–0.70] [–1.52] [2.66] |$R^2$| 0.83 0.78 0.81 Observations 205,822 177,311 28,511 The dependent variable in this table is yield spread in percentage. All regressions include three sets of fixed effects: time fixed effects, firm fixed effects, and ratings fixed effects (by notch). Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. K/V (in decimals) and Asset vol (in percentage) are calculated following Campbell, Hilscher, and Szilagyi (2008). Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. Amihud, Amihud vol, IRC, and IRC vol are the Amihud measure, the volatility of the Amihud measure, implied round-trip cost, and the volatility of implied round-trip cost, respectively. All four are calculated following Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab We next turn to hedge ratio in Table A.2. To allow for additional variation in hedge ratios, we interact ratings fixed effects with equity returns and also allow for interactions between equity returns and |$T$|⁠, |$T^2$|⁠, asset volatility, asset volatility|$^2$|⁠, |$\frac{K}{V}$|⁠, and |$\left(\frac{K}{V}\right)^2$|⁠. Even with these controls, we continue to find an economically and statistically significant effect of de facto seniority, though the magnitudes are somewhat smaller than in Table 8. Table A.2 Hedge ratios with nonlinear controls . Full . IG . Junk . |$r_E \times \text{Prop due prior}$| 0.0590 0.0556 0.0823 [2.89] [2.39] [4.15] |$r_E \times T$| 0.0034 0.0038 0.0011 [5.00] [4.52] [1.04] |$r_E \times T^2$| –0.0000 –0.0000 –0.0000 [–4.40] [–3.59] [–0.70] |$r_E \times \text{Asset vol}$| 0.0044 0.0045 0.0041 [1.78] [1.65] [1.35] |$r_E \times \text{Asset vol}^2$| –0.0001 –0.0001 –0.0001 [–1.88] [–1.72] [–1.57] |$r_E \times \text{K/V}$| 0.1069 0.0571 0.2576 [0.91] [0.53] [2.01] |$r_E \times \text{(K/V)}^2$| –0.0102 0.0554 –0.1484 [–0.12] [0.46] [–1.54] |$r_E \times \text{Firm T}$| 0.0060 0.0045 0.0114 [2.66] [2.13] [2.63] Prop due prior 0.0018 0.0018 0.0011 [1.77] [1.76] [0.71] |$T$| –0.0000 –0.0001 0.0000 [–0.71] [–0.74] [0.46] |$T^2$| 0.0000 0.0000 –0.0000 [0.88] [0.84] [–0.32] Asset vol 0.0003 0.0002 0.0006 [1.24] [0.98] [1.38] Asset vol|$^2$| 0.0000 0.0000 –0.0000 [0.91] [1.21] [–0.39] K/V 0.0210 0.0250 0.0223 [2.33] [2.51] [1.45] (K/V)|$^2$| 0.0037 0.0011 0.0009 [0.36] [0.08] [0.08] Firm T –0.0001 –0.0001 –0.0002 [–0.69] [–0.60] [–0.44] |$R^2$| 0.32 0.34 0.28 Observations 238,449 207,067 31,382 . Full . IG . Junk . |$r_E \times \text{Prop due prior}$| 0.0590 0.0556 0.0823 [2.89] [2.39] [4.15] |$r_E \times T$| 0.0034 0.0038 0.0011 [5.00] [4.52] [1.04] |$r_E \times T^2$| –0.0000 –0.0000 –0.0000 [–4.40] [–3.59] [–0.70] |$r_E \times \text{Asset vol}$| 0.0044 0.0045 0.0041 [1.78] [1.65] [1.35] |$r_E \times \text{Asset vol}^2$| –0.0001 –0.0001 –0.0001 [–1.88] [–1.72] [–1.57] |$r_E \times \text{K/V}$| 0.1069 0.0571 0.2576 [0.91] [0.53] [2.01] |$r_E \times \text{(K/V)}^2$| –0.0102 0.0554 –0.1484 [–0.12] [0.46] [–1.54] |$r_E \times \text{Firm T}$| 0.0060 0.0045 0.0114 [2.66] [2.13] [2.63] Prop due prior 0.0018 0.0018 0.0011 [1.77] [1.76] [0.71] |$T$| –0.0000 –0.0001 0.0000 [–0.71] [–0.74] [0.46] |$T^2$| 0.0000 0.0000 –0.0000 [0.88] [0.84] [–0.32] Asset vol 0.0003 0.0002 0.0006 [1.24] [0.98] [1.38] Asset vol|$^2$| 0.0000 0.0000 –0.0000 [0.91] [1.21] [–0.39] K/V 0.0210 0.0250 0.0223 [2.33] [2.51] [1.45] (K/V)|$^2$| 0.0037 0.0011 0.0009 [0.36] [0.08] [0.08] Firm T –0.0001 –0.0001 –0.0002 [–0.69] [–0.60] [–0.44] |$R^2$| 0.32 0.34 0.28 Observations 238,449 207,067 31,382 The dependent variable in this table is log corporate bond returns in decimals. |$r_E$| is the log equity return of the same issuer as the corporate bond and |$r_T$| is the log Treasury bond return of a Treasury bond with the same maturity as the corporate bond. Both |$r_E$| and |$r_T$| are expressed in decimals. Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. K/V (in decimals) and Asset vol (in percentage) are calculated following Campbell, Hilscher, and Szilagyi (2008). Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. All regressions contain firm fixed effects, ratings fixed effects (by notch), firm fixed effects interacted with |$r_E$|⁠, and ratings fixed effects interacted with |$r_E$| and |$r_T$| separately. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab Table A.2 Hedge ratios with nonlinear controls . Full . IG . Junk . |$r_E \times \text{Prop due prior}$| 0.0590 0.0556 0.0823 [2.89] [2.39] [4.15] |$r_E \times T$| 0.0034 0.0038 0.0011 [5.00] [4.52] [1.04] |$r_E \times T^2$| –0.0000 –0.0000 –0.0000 [–4.40] [–3.59] [–0.70] |$r_E \times \text{Asset vol}$| 0.0044 0.0045 0.0041 [1.78] [1.65] [1.35] |$r_E \times \text{Asset vol}^2$| –0.0001 –0.0001 –0.0001 [–1.88] [–1.72] [–1.57] |$r_E \times \text{K/V}$| 0.1069 0.0571 0.2576 [0.91] [0.53] [2.01] |$r_E \times \text{(K/V)}^2$| –0.0102 0.0554 –0.1484 [–0.12] [0.46] [–1.54] |$r_E \times \text{Firm T}$| 0.0060 0.0045 0.0114 [2.66] [2.13] [2.63] Prop due prior 0.0018 0.0018 0.0011 [1.77] [1.76] [0.71] |$T$| –0.0000 –0.0001 0.0000 [–0.71] [–0.74] [0.46] |$T^2$| 0.0000 0.0000 –0.0000 [0.88] [0.84] [–0.32] Asset vol 0.0003 0.0002 0.0006 [1.24] [0.98] [1.38] Asset vol|$^2$| 0.0000 0.0000 –0.0000 [0.91] [1.21] [–0.39] K/V 0.0210 0.0250 0.0223 [2.33] [2.51] [1.45] (K/V)|$^2$| 0.0037 0.0011 0.0009 [0.36] [0.08] [0.08] Firm T –0.0001 –0.0001 –0.0002 [–0.69] [–0.60] [–0.44] |$R^2$| 0.32 0.34 0.28 Observations 238,449 207,067 31,382 . Full . IG . Junk . |$r_E \times \text{Prop due prior}$| 0.0590 0.0556 0.0823 [2.89] [2.39] [4.15] |$r_E \times T$| 0.0034 0.0038 0.0011 [5.00] [4.52] [1.04] |$r_E \times T^2$| –0.0000 –0.0000 –0.0000 [–4.40] [–3.59] [–0.70] |$r_E \times \text{Asset vol}$| 0.0044 0.0045 0.0041 [1.78] [1.65] [1.35] |$r_E \times \text{Asset vol}^2$| –0.0001 –0.0001 –0.0001 [–1.88] [–1.72] [–1.57] |$r_E \times \text{K/V}$| 0.1069 0.0571 0.2576 [0.91] [0.53] [2.01] |$r_E \times \text{(K/V)}^2$| –0.0102 0.0554 –0.1484 [–0.12] [0.46] [–1.54] |$r_E \times \text{Firm T}$| 0.0060 0.0045 0.0114 [2.66] [2.13] [2.63] Prop due prior 0.0018 0.0018 0.0011 [1.77] [1.76] [0.71] |$T$| –0.0000 –0.0001 0.0000 [–0.71] [–0.74] [0.46] |$T^2$| 0.0000 0.0000 –0.0000 [0.88] [0.84] [–0.32] Asset vol 0.0003 0.0002 0.0006 [1.24] [0.98] [1.38] Asset vol|$^2$| 0.0000 0.0000 –0.0000 [0.91] [1.21] [–0.39] K/V 0.0210 0.0250 0.0223 [2.33] [2.51] [1.45] (K/V)|$^2$| 0.0037 0.0011 0.0009 [0.36] [0.08] [0.08] Firm T –0.0001 –0.0001 –0.0002 [–0.69] [–0.60] [–0.44] |$R^2$| 0.32 0.34 0.28 Observations 238,449 207,067 31,382 The dependent variable in this table is log corporate bond returns in decimals. |$r_E$| is the log equity return of the same issuer as the corporate bond and |$r_T$| is the log Treasury bond return of a Treasury bond with the same maturity as the corporate bond. Both |$r_E$| and |$r_T$| are expressed in decimals. Proportion due prior is the proportion of the face value of an issuer’s debt due before a bond, expressed in decimals. T is the time-to-maturity in years. K/V (in decimals) and Asset vol (in percentage) are calculated following Campbell, Hilscher, and Szilagyi (2008). Firm T is the weighted-average time-to-maturity of all of the issuer’s bonds outstanding, weighted by amount outstanding. All regressions contain firm fixed effects, ratings fixed effects (by notch), firm fixed effects interacted with |$r_E$|⁠, and ratings fixed effects interacted with |$r_E$| and |$r_T$| separately. t-stats are in brackets and use standard errors two-way clustered by firm and time. Open in new tab D. Fitting the Model In Section 1, we examine the implications of the extended Merton model using a case in which 10% of a firm’s debt is due prior (10/10/80) and one where 80% of a firm’s debt is due prior (80/10/10). Here, we extend our numerical calculations to allow for 5%, 20%, 30%, 40%, 50%, 60%, 70%, and 85% of a firm’s debt to be due prior to the intermediate bond in addition to the 10% and 80% cases. For each proportion due prior, our goal is to calculate a surface that determines yield spreads and hedge ratios as a function of a firm’s asset volatility and market leverage. To do this, we perform numerical calculations over 546 grid points for each value of proportion due prior, intersecting asset volatilities that range from 2.5% to 65% and market leverage ranging from 2.5% to 90%. All of the grid points are also checked using simulations to verify accuracy. Once we have 546 grid points, each surface is generated by using Matlab to interpolate grid points. With the surfaces, we are able to calculate yield spreads and hedge ratios for |$(\sigma_v, \text{market leverage}, \text{proportion due prior})$| triplets. For each observation in our empirical sample, we determine its asset volatility using the calculations described in Equations (A.4) and (A.5) and its market leverage as (DLC |$+$| DLTT)/(DLC |$+$| DLTT |$+$| market equity). We then use the closest proportion due prior surface pair to calculate the marginal effect of yield spreads and hedge ratios for an observation, analogous to a numerical derivative. For example, the effect of de facto seniority estimated for a bond with a proportion due prior of 0.35 is estimated using the 30% and 40% proportion due prior grids as \begin{align} \frac{f(\sigma_v,\text{market leverage},0.4) - f(\sigma_v,\text{market leverage},0.3)}{0.4 - 0.3}, \end{align}(A.6) where f(.) is the function for the model yield spread or hedge ratio. Some of the results in this paper were previously reported in a working paper titled “Comovement of Corporate Bonds and Equities.” For helpful comments and discussions, we thank Robin Greenwood (the editor), Manuel Adelino, Geert Bekaert, Markus Brunnermeier, Sergey Chernenko, Alex Edmans, Jean Helwege, Jingzhi Huang, Rainer Jankowitsch, Adam Kolasinski, Martin Oehmke, Jun Pan, Eric Powers, Marco Rossi, Ilya Strebulaev, Rene Stulz, Mathijs van Dijk, Yuhang Xing, Wei Xiong, and Xing Zhou; two anonymous referees; seminar participants at Baruch, BI Norwegian Business School, Boston University, Dimensional Fund Advisors, Drexel, Erasmus, the Federal Reserve Board, INSEAD, Norges Bank Investment Management, North Carolina State, Ohio State, Rutgers, Seoul National, Texas A&M, University of Hong Kong, University of Illinois at Chicago, and University of South Carolina; and conference participants at the EFA Meetings, Inquire UK, and the South Carolina Fixed Income Conference. 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Although it is straightforward to extend the Merton model to allow for explicitly junior and senior bonds, it is difficult to empirically examine the effect of explicit seniority on corporate bond prices because most corporate bond issues in the United States are senior unsecured. Furthermore, though bank debt is often senior to corporate bonds, firms with significant amounts of corporate bonds outstanding often do not have significant bank debt. Rauh and Sufi (2010) report that more than 50% of the firms in their sample with significant amounts of corporate bonds have less than 10% of their debt in bank debt. Our sample, which is at the bond-month level, is even more extreme, with a median of approximately 2% of debt in bank debt. 3 As another benchmark, consider that Dick-Nielsen, Feldhutter, and Lando (2012) find a 1-4 basis point liquidity effect precrisis and a 5-92 basis point liquidity effect post-crisis for investment-grade bonds. Thus, a 48 basis point effect for a bond’s place in its firm’s maturity structure is economically large. 4 Explicit seniority structure is considered by Moody’s when rating bonds (Fons et al. 2007), but de facto seniority is not. In our sample, a regression of ratings on proportion due prior (our measure of de facto seniority) and firm fixed effects has a within group |$R^2$| of only 0.0001. 5 In a recent paper, Feldhutter and Schaefer (2015) use a long history to estimate default probabilities and find a smaller credit spread puzzle. Another facet of the credit spread puzzle, the difficulty in explaining changes in yield spreads is illustrated by Collin-Dufresne, Goldstein, and Martin (2001). Other relevant papers focused on understanding yield spreads, returns, and their moments include Schaefer and Strebulaev (2008), who focus on hedge ratios and Campbell and Taksler (2003), Campbell and Taksler (2003), and Bao et al. (2014), who focus on yield spreads and volatilities. There is also an extensive literature focusing on the comovement between Treasury bonds and equities. Recent papers include Baele, Bekaert, and Inghelbrecht (2010), Baker and Wurgler (2012), and Campbell, Pflueger, and Viciera (2013). In addition, a line of literature examines bond and equity returns over corporate announcement windows, including Hotchkiss and Ronen (2002), Maxwell and Stephens (2003), and Maxwell and Rao (2003). 6 See also Chen, Xu, and Yang (2012), and He and Milbradt (2016). 7 Stochastic interest rates, an important source of variation in bond prices, are not modeled here as the tie between bonds and equities through interest rates is indirect and less important than the tie through firm value. See Shimko, Tejima, and van Deventer (1993) for an extension of the Merton model to stochastic interest rates and Schaefer and Strebulaev (2008) and Campbell and Taksler (2003) for applications of this model. 8Brunnermeier and Oehmke (2013) use similar intuition in their theoretical analysis to motivate an incentive to shorten maturities, which, in turn, leads to a maturity rat race for financial firms. They posit that nonfinancial firms are better able to commit to a maturity structure and avoid a rat race. 9 If the firm is solvent at |$T_i$|⁠, firm value drops to |$V_i - K_i$| and the firm value process follows a geometric Brownian motion between |$T_i$| and |$T_{i+1}$|⁠. 10 We do not formally model the decision to rollover debt, but the states where |$V_i$| is barely larger than |$K_i$| are the states where the rollover of debt would be particularly expensive, if not impossible. These states are primarily responsible for driving the wedge between bond issues that creates de facto seniority. See He and Xiong (2012) and He and Milbradt (2014) for recent papers that examine the issue of rollover risk. 11 The way that we model default is akin to assuming that all defaults are resolved in bankruptcy courts. We abstract away from complexities such as out-of-court settlements and exchange offers where some bondholders may receive larger recoveries than other bondholders. Firms that decide to try to avoid bankruptcy by making exchange offers have incentives to make these exchange offers to debt that is maturing soon since this is the debt for which they will have to service a significant face value in the near future. To the extent that firms need to make favorable offers to bondholders to convince bondholders to accept exchange offers, this also makes earlier maturing debt de facto senior. 12 Our set-up essentially maps all bonds maturing prior to |$T_2$| into a single bond that matures at |$T_1$| and all bonds maturing after |$T_2$| into a single bond that matures at |$T_3$|⁠. 13 The maturity |$T$|⁠, in the Merton model, is set to 7 years to be in line with typical medium-term bonds. 14 The proportion of debt due prior to the intermediate bond (10% in the first scenario and 80% in the second scenario) thus measures how de facto senior the intermediate bond is in these numerical examples. We will use the same variable to measure the de facto seniority of a bond in our empirical tests. 15 Within the model, this can be thought of as either |$V_1 - K_1 < K_2$|⁠, in which case the repayment of the first bond has already driven firm value below what |$T_2$| debtholders are owed or a case in which |$V_1 - K_1 > K_2$|⁠, but not by that much. In such a case, |$T_2$| debt has a high probability of not being fully paid if firm performance between |$T_1$| and |$T_2$| is poor. 16 Another possibility is that bankruptcy courts treat bonds with different maturities as different classes in the resolution of bankruptcy. Guha (2002) notes that in most Chapter 11 cases, bonds with the same explicit seniority are put in the same class. But to the extent that bonds are occasionally put into different classes by maturity, this could also generate a de facto seniority effect. 17 Bonds where the only provision is a make whole call provision, which constitutes a significant portion of nonfinancial bonds, are kept. Make whole calls involve redemption at the maximum of par and the present value of all future cash flows using a discount rate of a comparable Treasury plus X basis points, where the most common values of X are 20, 25, and 50. Historically, the average AAA yield spread has been greater than 50 basis points, leaving make whole calls with little chance of being in-the-money. Powers and Tsyplakov (2008) show that theoretically, make whole calls have little effect on bond prices. They also find that by 2002-2004, the empirical effect of make whole calls on bond prices is extremely small. 18 Though some corporate bond studies, primarily those on corporate bond illiquidity, retain financial firms, studies on structural models of default typically drop financial firms because leverage has different implications for financial firms. Furthermore, Brunnermeier and Oehmke (2013) argue that financial firms have incentives to enter a maturity rat race and are unable to commit to a maturity structure. Thus, any measure of maturity structure today for a financial firm is unlikely to reflect bondholders’ expectations of future maturity structure. 19 We also apply standard filters for corrected and canceled trades and eliminate obvious errors. 20 We also require trades to be at least 11 business days apart and no more than 31 business days apart to calculate returns as trades that deviate too much from being monthly could be nonrepresentative. For example, if a bond’s consecutive last trades of the month are on the last day of the first month and the first day of the second month, we would be calculating a daily return and scaling up to a monthly return. This may create an outlier. 21 Note that a firm-month can be represented in both the investment-grade subsample and the speculative-grade subsample if it has some bonds with investment-grade ratings and some bonds with speculative-grade ratings. 22 Directly using market equity from CRSP for May 2010 introduces errors if the firm had significant issuance or repurchases between December 2009 and May 2010 that would be reflected in the market equity for May 2010, but not in the book equity for December 2009. 23 In our sample, the correlation between proportion due prior and T is high at 0.73. We run variance inflation factor (VIF) tests following Kennedy (1998), and find VIFs under four for all of our yield spreads specifications, far under the benchmark of 10 at which point multicollinearity may significantly affect statistical inference. 24 We thank an anonymous referee for suggesting this test. 25 We include differences in ratings as a control where a positive difference indicates that bond |$j$| has a poorer credit rating. For the vast majority of cases, the difference in ratings is zero. 26 We separately sort the panel by Amihud and IRC and retain only pairs for which both measures are within 20 percentile points. 27 An alternative would be to simply scale realized corporate bond returns by realized equity returns. Such an estimate would be very imprecise as cases of small equity returns relative to large corporate bond returns would create extreme observations. 28 In these hedge ratio regressions, |$r_E \times \text{proportion due prior}$| is highly correlated with both |$r_E$| and |$r_E \times T$|⁠, which raises the issue of multicollinearity. Variance inflation factor tests confirm this potential concern. To diagnose the possibility that our estimates are significantly affected by multicollinearity, we use the fact that multicollinearity has the symptom of producing wide swings in parameter estimates with even small changes in data (Greene 2003). We randomly drop half of our observations and reestimate our main hedge ratio specification in 1,000 simulations. We find that our results are stable, inconsistent with a significant multicollinearity effect. 29 As our focus is on equity hedge ratios, we allow the equity hedge ratio to vary with firm characteristics. Potentially, Treasury hedge ratios can also vary with these same characteristics. As a robustness check, we add |$r_T\times \text{BE/ME}$| and |$r_T \times \text{Equity vol}$| to our specifications. Furthermore, we include |$r_T \times\text{proportion due prior}$|⁠. The coefficient on |$r_E\times\text{proportion due prior}$| is 0.0850 (⁠|$t$|-stat |$=$| 5.13), little changed from our base specifications. The coefficient on |$r_T\times\text{proportion due prior}$| is |$-$|0.1540 (⁠|$t$|-stat = |$-$|3.48), again consistent with the fact that bonds due later in their issuer’s maturity structure are more equity-like and less Treasury-like. The coefficients on |$r_T\times \text{BE/ME}$| and |$r_T \times \text{Equity vol}$| are both negative and statistically significant, consistent with Treasury hedge ratios being lower for firms with greater credit risk. 30 This contrasts our findings for yield spreads where the de facto seniority effect is stronger for the speculative-grade subsample. We investigate this difference in Section 3.3. 31 Further adding |$r_E\times\text{Amihud vol}$| and |$r_E\times\text{IRC vol}$| has little effect on our results. 32 The only other covenants in at least 20% of our sample are cross acceleration, change of control puts (related to consolidation/merger covenants), and sales leaseback (similar to negative pledge covenants in preventing what is economically similar to secured debt). 33 The empirical effect is |$-$|2 basis points with a |$t$|-stat of |$-$|0.08. 34Masulis and Korwar (1986) find that in a sample of 1,406 announcements of stock offerings, 372 offerings are planned exclusively to refund outstanding debt. This suggests that there is a realistic role for equityholders in paying off debt. 35 See also Leland (1994) and Leland and Toft (1996) for models that incorporate equityholders’ decisions on whether or not to service debt. We discuss the Leland-Toft model in more detail in Appendix B.3. 36 The slightly lower yield spreads and hedge ratios for the case in which the bond is relatively late in the maturity structure is attributable to the fact that we model bonds as zero-coupon bonds and the recovery as being in proportion to the face value. Thus, bonds late in the maturity structure receive a slightly disproportionate recovery relative to market values. Published by Oxford University Press on behalf of The Society for Financial Studies 2017. This work is written by US Government employees and is in the public domain in the US. Published by Oxford University Press on behalf of The Society for Financial Studies 2017.
TI - De Facto Seniority, Credit Risk, and Corporate Bond Prices
JF - The Review of Financial Studies
DO - 10.1093/rfs/hhx082
DA - 2017-11-01
UR - https://www.deepdyve.com/lp/oxford-university-press/de-facto-seniority-credit-risk-and-corporate-bond-prices-zhLp3x1Hu3
SP - 4038
EP - 4080
VL - 30
IS - 11
DP - DeepDyve
ER -