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Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds∗ Stephen M. Schaefer Ilya A. Strebulaev London Business School This version: 10 November 2003 ABSTRACT It is well known that structural models of credit risk provide poor predictions of bond prices. We show that they may perform much better as a predictor of debt return sensitivities to equity. This is important since it gives us an opportunity to identify much better the reasons for model failure. The main result of this paper is that even the simplest of the structural models (Merton (1974)) produces hedge ratios that are in line with those observed empirically. As well as providing insight into the determinants of corporate bond prices our results are also useful to practitioners who wish to hedge their positions in corporate debt. The paper also shows that corporate bond prices are sensitive to some variables – e.g., VIX – in a way that appears unrelated to credit risk. Keywords: Credit risk, structural models, hedge ratios, credit spreads JEL Classification Numbers: G12, G13 ∗ We would like to thank Crispin Southgate and Joseph Nehoraj from Merrill Lynch and European Credit Management for help with the data. We are responsible for all remaining errors. Address for correspondence: Institute of Finance and Accounting, London Business School, Regent’s Park, London NW1 4SA, UK. E-mail: [email protected]. Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds Abstract It is well known that structural models of credit risk provide poor predictions of bond prices. We show that they may perform much better as a predictor of debt return sensitivities to equity. This is important since it gives us an opportunity to identify much better the reasons for model failure. The main result of this paper is that even the simplest of the structural models (Merton (1974)) produces hedge ratios that are in line with those observed empirically. As well as providing insight into the determinants of corporate bond prices our results are also useful to practitioners who wish to hedge their positions in corporate debt. The paper also shows that corporate bond prices are sensitive to some variables – e.g., VIX – in a way that appears unrelated to credit risk. Keywords: Credit risk, structural models, hedge ratios, credit spreads. I Introduction It is commonly agreed that structural models of credit risk over-value corporate bonds.1 Structural models employ the contingent claims approach to value the put option inherent in the contract between lenders and equityholders. Contingent claims models are, of course, widely used in practice and are seen as one of the major successes of financial theory. The failure of such models to explain satisfactorily actual corporate debt prices and spreads is therefore surprising. The poor performance of these models in this area has been recognized for many years but their failure continues to surprise. This paper makes two simple but important points. First, while structural models provide a poor prediction of prices and returns, they may perform much better as a predictor of the sensitivity – or hedge ratio – of debt to equity. This is important because hedge ratios determine the composition of the replicating portfolio which, according to the theory, determines the price. Thus, if we find that a model provides a good prediction of hedge ratios but a poor prediction of the price, we are better able to identify the reasons for model failure. In fact, we find that even the simplest structural model (Merton (1974)) predicts hedge ratios that are in line with those observed empirically. This leads us to reconsider the possible explanations for the failure of structural models to explain better the level of prices and yields. At present, there are two main explanations for this failure. First, that structural models fail to predict accurately the probability of default and/or the recovery rate. This explanation has weight because all current models violate at least some known facts about capital structure and/or the circumstances of corporate default. Consequently, the development of much of the theory has been in pursuit of improved ways to model credit events. As a result, we possess an arsenal of models that include stochastic default boundaries, dynamic capital structure and opportunistic behavior on the part 1 For early empirical investigation of the Merton model see Jones, Mason, and Rosenfeld (1984). For a more recent analysis see Eom, Helwege and Huang (2003) who study the empirical performance of a number of structural models. 2 of claimholders (e.g., Anderson and Sundaresan (1996), Collin-Dufresne and Goldstein (2001)). Empirical tests have showed, however, that these modifications do not substantially improve the ability of the models to explain the level of corporate bond prices (Eom, Helwege and Huang (2002), Huang and Huang (20002)). A second explanation is that the pricing of corporate bonds is influenced by factors that are not related to credit risk, and are therefore not included in structural models. It could even be the case that structural models account very well for the credit risk component of bond prices and returns while, at the same time, credit risk is actually responsible for only a part, perhaps not even a very large part, of spreads and returns. Some recent results tend to support this view. Elton, Gruber, Agrawal, and Mann (2001) find that differences in taxation account for about a third of credit spreads. Huang and Huang (2002) estimate that credit risk accounts only for a small fraction of the observed credit spread. Collin-Dufresne, Goldstein, and Martin (2001) find that the variables present in structural models can not explain changes in spreads. On the positive side, a recent paper by Leland (2002) shows that the default probability prediction of structural models are indeed roughly consistent with observed default frequency.2,3 Our paper contributes to this on-going debate but, unlike many previous authors (e.g., Huang and Huang (2002)), we do not focus on the level of prices or the size of the spread. Instead we investigate the ability of structural models to explain rates of return and we ask is whether these models can be used to hedge corporate bond returns. Using data on monthly returns for a large sample of U.S. corporate bonds over a five-year period, we find that the variables present in structural models explain a large fraction of the returns on investment grade bonds and a smaller but significant fraction for high yield bonds. This result is in itself not surprising, since it has been known for some 2 This finding means that those models which try to find the ways to make predicted spreads closer to the actual ones, can substantially overpredict default frequency. 3 This result is also in line with the eponymous KMV method that used a Merton-based approach to predict “distance to default”. 3 time that investment grade and government bonds have similar returns (Campbell and Ammer, 1993). We also find that debt returns are significantly related to returns on the underlying asset and that the pattern of sensitivities is broadly consistent with the level of credit exposure. In other words, this paper focusses attention on the second-moment predictions of the model. In structural models, any change in the value of a credit risky bond credit is a result of a change in the value of the assets that collateralize the debt or a change in riskless rates. In our empirical analysis we employ the change in the value of the equity together with descriptors of the change in riskless rates and ask whether the sensitivities of corporate bond returns to equity and riskless bond returns are consistent with the model. Our main result is that even the Merton (1974) model produces equity sensitivities that are roughly in line with those observed empirically. Our test is supportive of the view that structural models account well for the credit risk component of corporate bond returns. This positive result is also consistent with Leland’s favorable and recent findings on the default probability predictions of structural models. A number of authors have found that the returns on corporate bonds are also related to a number of factors that are not present in structural models. These include the FamaFrench SMB and HML factors (Elton et. al. (2001)), returns on a broad index of equity prices and implied volatility from options on equity indices, “VIX” (Collin-Dufresne et. al. (2001)). In the second half of the paper we include these variables in our analysis of hedged corporate bond returns. The results represent the second main finding of this paper. We find that returns on corporate bonds are significantly related to changes in the VIX implied volatility index but that the sensitivities are not related to any of the standard measures of credit exposure such as rating, leverage or asset volatility. Our results on the Fama-French SMB factor are similar. Thus VIX and SMB have significant effects on the prices of corporate bonds but are not related to their credit risk.. Our results also have potential interest for practitioners who wish to hedge corporate 4 debt positions. Our findings suggest that structural models are in fact more useful for this purpose than might appear from their performance in explaining the size of credit spreads. Other authors have previously studied these issues. Blume, Keim, and Patel (1991) also study the behavior of corporate bond returns but not within the framework of structural models. Huang and Huang (2002) use a variety of models to determine whether structural models explain the average level of yield spreads but do not study hedge ratios. Collin-Dufresne, Goldstein, and Martin (2001) analyze changes in yield spreads in a regression framework where the choice of regressors is motivated by structural models. They do not, however, examine whether the size (as distinct from the sign) of the estimated coefficients is consistent with the theory. The closest paper to ours in spirit is Leland (2002) in sense that we investigate the predictions of structural models that do not directly touch on the level of bond prices (or yield spreads). While Leland (2002) studies whether structural models are able to predict the presence of corporate default, we study their implications for hedge ratios and so attempt to define more narrowly the reasons why these models fail to explain the level of spreads. Our paper proceeds as follows. Section II provides a description of the data set and the sample selection procedure and also gives descriptive statistics. Section III describes some preliminary regression analysis of ability of structural models to explain returns on corporate bonds. In section IV the procedure is refined and the regressions take into account the effect of changes in asset values, volatility and leverage on the hedge ratios predicted by the structural model. Section V examines the sensitivity of bond returns to other variables such as VIX and the Fama-French SMB and HML factors. Section VI concludes. 5 II Data, Sample Selection and Descriptive Statistics II.1 Data We use monthly prices on corporate bonds that are included either in the Merrill Lynch Corporate Master index or the Merrill Lynch Corporate High Yield index. These indices include most rated U.S. publicly issued corporate bonds. The data covers the period from December 1996 to September 2002. Table I provides descriptive statistics on the bonds in the data set. The data set contains more than 323000 bond-month observations, with about 2900 issuers and 9000 issues. Matching with CRSP and COMPUSTAT allows us to use about 50% of the total number of observations and all rating categories (from AAA to CCC) are represented. As we would expect, as we move down the ratings the average time-to-maturity decreases and the average coupon rate increases. The median size at issuance is $200 million dollars. Detailed information on each bond is obtained from the Fixed Income Securities Database (FISD) as provided by LJS Global Services) and equity and treasury bond returns are from CRSP. II.2 Sample selection The specific bonds included in the analysis satisfy the following criteria: (1) the bond is not convertible, exchangeable, callable or putable; (2) the bond is issued by a U.S. company and denominated in $U.S.;4 (3) it is possible to match unambiguously the bond issuer with a company in CRSP using the CUSIP; (4) the bond is issued by a nonfinancial corporation; (5) the bond has an initial maturity of at least four years and (6) the bond has at least 25 consecutive monthly price observations. Table I gives summary statistics for the entire sample and Table II for the remaining sample of 1362 bonds. II.3 Descriptive statistics on returns For each bond j we calculate the return between months t and t − 1 as follows: 4 More specific, the company is of the U.S. origin according to the FISD definition. In particular, its headquarters should be located in the U.S. and it is subject to the U.S. legal practice. 6 rj,t = Pj,t + AIj,t + Ij,t Cj /Nj , Pj,t−1 where Pj,t is the price of bond j at the end of month t and AIj,t is the change in the accrued interest between t − 1 and t. Since the calculation of the accrued interest restarts with each coupon payment, if the coupon date falls between t − 1 and t, Cj /Nj is added to the price, where Cj is the annual coupon rate and Nj is the coupon frequency per annum of bond j. Ij,t is an indicator function taking the value of 1 if the coupon is due between t − 1 and t. The excess return is then calculated as rj,t = rj,t − rf1m,t , where rf1m,t is the return on one-month Treasury bills up to month t. Table III provides summary statistics on individual raw returns and excess returns for the whole data set and by rating (where rating, as provided by the S&P, is defined as the rating on the first date that a bond is present in the dataset). The realized average monthly return on corporate bonds is 0.50% and the average excess return is 0.11%.5 Over this period the low-grade bonds had a lower average return than investment-grade bond and a substantially higher standard deviation of rate of return. for the period 1977-1989. III Structural models and returns In this section we study the determinants of corporate bond returns implied by structural credit risk models. We take initially a conservative view and include in the the regressions only those factors which are explicitly stochastic in the Merton model. Later other variables are included – specifically VIX and the Fama-French factors – and the sensitivity of debt returns to equity re-examined. 5 We include these statistics on mean returns only for comparison. The period covered by our data is too short to be able to identify means with any precision. The focus of our study, however, is on second moments (e.g. hedge ratios) and for this purpose the data is adequate. 7 The value of the firm, V , is the driving state variable in most structural models; indeed the presence of V is the distinguishing feature of the structural approach. In this section we estimate the sensitivity of debt values to changes in the value of the firm by regressing excess rates of return on corporate bonds against the excess return on the equity of the issuing firm.6 In a one-factor model the elasticity of the value of debt to equity is related to the delta of the debt against V by: ∂D E = ∂E D à ∂D ∂V ∂E ∂V ! E = D à 1 ∂E ∂V ! −1 E , D (1) Clearly, this elasticity is a function of both V and interest rates and therefore varies over time. These regressions are, therefore, simply a first step. Later in the paper we use the Merton model to account for time variation in the elasticity. In our regressions we also control for changes in the riskless term structure by including returns on a ten-year constant maturity Treasury bond. While in the first generation of structural models the risk-free rate was held constant, more recent versions include a stochastic interest rate (e.g., Kim, Ramaswamy, and Sundaresan (1993), Longstaff and Schwartz (1995)). An increase in the risk-free rate has two opposing effects on the price of debt. First, it decreases the value of debt by decreasing the present value of all future cash flows. Second, it leads to increase in the risk-neutral drift of the value or the firm, V , and so increases the value of debt by decreasing the likelihood of default. The second effect will be relatively more important for bonds with a higher likelihood of default. Thus we regress the excess return on each bond, rj,t , on the excess return on the issuing firm’s equity, rE,t , and return on riskless bonds, rf 10y,t : rj,t = αj,0 + αj,E rE,t + αj,rf rf 10y,t , 6 (2) An increase in E causes an increase in D in case this increase is caused by an increase in V , i.e. an increase in the total pie to be divided between equityholders and debtholders. It also can be the case that an increase in E is a result of a wealth transfer from debtholders to equityholders (with V kept constant). The latter issue has been investigated in the structural framework by the strategic debt service models such as Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1997). 8 Table IV reports the average value of the coefficients and their t-statistics calculated from their cross-sectional standard deviation.7 A number of points are worth noting. Both factors are significant for the whole sample and for most of the rating categories. For the whole sample, a one percent return on the riskless bond leads to 0.45% increase in the corporate debt price. The standard deviation of treasury bond returns is about 1.5% per month and so the average impact on one-month returns on corporate bonds of a one standard deviation return on government bonds is about 0.75%. The impact of the riskless rate becomes smaller for lower credit rating categories: the loading is significantly positive for investment-grade bonds and negative but insignificant for B and CCC grade bonds. This suggests, quite interestingly, that for very low quality bonds the effect of an increase in the riskless rate – and thus the risk-neutral drift of V – in reducing credit exposure may outweigh the resulting decline in the present value of firm’s liabilities. Ignoring the effects of credit risk and assuming parallel shifts in the riskless term structure, the coefficient on the return on Treasuries would equal the ratio of the duration of the corporate bond to the duration of the Treasury. As the average maturity of our sample of corporate bonds is just under 13.5 years (see Table II) and the Treasuries have a constant maturity of 10 years, the actual coefficient of 0.75 is clearly lower than the ratio of the durations. This is consistent with a negative correlation between changes in the yield spreads on corporate debt and changes in Treasury yields. For the full sample, the return impact of a 1% return on equity is 0.042%. If we assume that the standard deviation of monthly equity returns is 12%, then a one standard deviation return on equity increases a bond’s return by 0.4% on average. This is smaller than, but of the same order of magnitude as the effect of the risk-free rate. The sensitivities of bond returns to equity are convex in the credit rating: a one percent increase in the stock price increases returns by 1-2 basis points (bp) for AA-A bonds, 4 7 Similar procedure was employed by Collin-Dufresne, Goldstein, and Martin (2001) in the context of studying the changes in credit spreads. 9 bp for BBB bonds, and 7-11 bp for BB-B bonds. The average coefficient on equity is significant for each of the ratings (apart from AAA).8 In univariate regressions (not reported here) returns on the riskless bond are the major determinant of returns on high-grade bonds, while for low-grade bonds equity is the more important. The two factors combined explain about half of a variation in returns on AAA-A-grade bonds and 20% for low-grade bonds. We show later that, even when other regressors are included and the results are computed for subperiods,, the sensitivity of corporate bond returns to both equity and riskless bonds remain significant and the coefficients exhibit the same relationship with credit ratings. IV IV.1 Debt sensitivity to equity Sensitivity in structural models Having established that the sensitivity of corporate debt returns to the underlying equity and riskless debt are significant, we now ask whether the magnitude of these sensitivities is consistent with the predictions of structural models. In this paper we employ the Merton (1974) model. This may be a surprising choice because, not only are its assumptions regarding capital structure clearly oversimplified, but it is also well known to underestimate credit spreads by a wide margin (Jones, Mason, and Rosenfeld, 1984). However, it remains an open question whether this simple model is able to explain the sensitivity of debt returns to equity and this is the question we now address. In the Merton model the value of equity is simply the value of a European call option on the assets of the firm with exercise price equal to the face value of the debt. Using equation (1) we may therefore write the sensitivity of the return on a credit risky bond to the return on equity as: 8 This is a different result from Collin-Dufresne, Goldstein, and Martin (2001) who find that changes in equity and quasi-market leverage do not have a significant impact on changes in credit spreads. 10 µ hE = 1 −1 ∆E ¶µ ¶ 1 −1 , L (3) where L is market leverage (defined as market value of debt as a percentage of the market value of the firm) and ∆E is the delta of a European call option on the value of the firm (equity). In the Merton model the parameters of interest are book leverage, B, the volatility of the firm’s assets, σA , time to maturity, T − t, and the risk-free rate, r. The table below shows the comparative statics of hE (assuming that the option to default is out-of-the-money). Parameter hE B + σA + T −t +/− r − For short maturity bonds the effect of an increase in time to maturity on the dispersion of asset values at maturity is greater than the effect on the risk-neutral expected value of the firm’s assets and the hedge ratio increases. For long maturity bonds the second effect dominates and an increase in time to maturity decreases the hedge ratio. Our calculations suggest that, for reasonable parameter values, the actual times-to-maturity in our sample are too small for the second effect to dominate. Table V shows the values of hE for the Merton model for asset volatilities between 10% and 50% and values of L between 10% and 70%. The risk-free rate is 5 %per annum and the time-to-maturity is 10 years. These values have been chosen to reflect those encountered in our sample. The table shows that the sensitivity varies significantly from zero (less than 0.01), when both leverage and volatility are low, to about 0.3 when both leverage and volatility are high. While in the remainder of the paper we focus on the Merton model, it would be straightforward to carry out the same analysis using other debt pricing models. To 11 illustrate, we consider two other models of corporate debt pricing. In the Leland (1994) model, where debt is assumed to be perpetual and to pay a constant coupon, the bond price is given by: " µ ¶−x # µ ¶−x Vt cF Vt ∗ 1− Pt = + max(wV − K, 0) , ∗ r V V∗ where c is the coupon rate, F is the principal, K and w are fixed and proportional liquidation (bankruptcy) costs, V ∗ is the time-invariant default boundary, µ ∗ V = cF r ¶ 1 , 1 + x−1 and x is constant and a function of other parameters, x = x(σ, δ, r), where δ is the firm’s payout ratio. Taking the derivative of Pt with respect to Vt , we find that ∂Pt = ∂Vt µ 1 V∗ ¶−x · Vt−x−1 x ¸ cF ∗ − max(wV − K, 0) . r Substituting the above formulas into equation (3) we obtain the corresponding value of hE . In the strategic debt service models of Anderson and Sundaresan (1996) and MellaBarral and Perraudin(1997), and if we also assume that the bond is a constant coupon perpetuity, the result is similar to Leland’s, the only difference being the value of the boundary: V∗ = IV.2 cF/r + K w (1 + x−1 ) Preliminary comparison of sensitivity in the Merton model and actual data As a first step in investigating the relationship between the Merton model sensitivities and those estimated empirically using regressions we perform a simulation. The object of this simulation is to calculate the mean value of the sensitivity of bond returns to equity in our data when (a) the Merton model holds and (b) the sensitivities are estimated 12 using linear regression on monthly data that has the same characteristics as our sample. For a given rating class, the difference between the mean sensitivity we obtain in this way and the sensitivity calculated from Merton model using the mean values by rating class, leverage, volatility, and time to maturity can be attributed to (a) non-linearity and (b) the discrete time interval between observations. We aggregate the seven principal rating classes into three: “AAA-A” (including AAA, AA, and A bonds), “BBB”, and “Junk” class (BB, B, and CCC bonds). For each of the three classes we find the 5% and 95% quantiles of leverage,9 volatility, and time to maturity and we take these to be the minimum and maximum points of unform distributions from which we draw values in the simulation.10 The table below gives the upper and lower limits for the three variables for each rating class. Parameter Leverage AAA-As BBB Junk min max 0.15 0.35 0.2 0.45 0.3 0.7 min max 0.1 0.3 0.15 0.35 0.2 0.5 min max 5 20 0.48 5 20 0.35 5 20 0.17 Asset volatility Time to maturity Frequency We now generate 2000 time series of “bond returns” as follows. First, we assign a rating class to each time series according to the proportions found in the actual data. Second, again for each time series, we randomly draw values for leverage, volatility, and time to maturity from the distributions for the relevant credit class and generate a time series of asset values. Using the Merton model we then calculate monthly equity and 9 10 The ratio of the face value of the firm’s debt to the market value of assets. We also tried to account better for the distribution patterns of leverage and volatility within the rating class; it did not produces any significant changes. 13 bond prices and, from these, monthly returns. Finally, we estimate the hedge ratio, hE , by running a regression of the simulated bond returns on simulated equity returns (i.e., a regression similar to (2) but excluding the return on the riskless bond.11 ) Table VI reports the mean values of hE and R̄2 for these regressions. Comparing these results and those in Table IV we see that the sensitivities are surprisingly similar. For high quality (AAA-A) bonds, the average sensitivity is found to be about 0.01 for simulations and 0.0003–0.02 for the actual the data; for BBB bonds both for the model and the actual data produce a sensitivity of 0.04 and for junk bonds we find 0.15 for the simulations and 0.07–0.11 bp for the actual data. The mean values of the simulated and empirical hedge ratios are not significantly different at a 5% confidence for the entire sample and the “AAA-A” and “BBB” subsamples and at the 10% level for the “Junk” subsample. These results are surprising since, for the same simulated data, the model underestimates the observed level of credit spreads by more than 80%, or by more than 50bp in absolute terms. IV.3 Preliminary Analysis of Hedge Ratios The results in the previous section raise the possibility that, although the Merton model leads to poor predictions of credit spreads, it may perform better as a predictor of hedge ratios. This result, if substantiated, would be important because, in contingent claims pricing theory, the hedge ratios define the composition of the replicating portfolio which, in turn, defines price of the contingent claim. Thus, if we find that the model provides good predictions of hedge ratios but poor predictions of the bond price, we are better able to identify the reasons for model failure. To address this issue, we next test the second-moment prediction of the model in a more rigorous manner. Note that if the model is correct then in equation (2) αj,E is 11 We also run regressions on the actual data using only equity returns to be consistent. The results are broadly similar with the equity sensitivities slightly larger. 14 equal to hE and we can therefore rewrite the regression as rj,t = αj,0 + βj,E hE,j,t rE,t + αj,rf rf 10y,t , (4) where hE,j,t is the model hedge ratio for firm j at time t and under the null hypothesis that the Merton model holds, βj,E is equal to one. To implement (2) the following parameters need to be estimated for each firm: the ratio of the book value of debt to the market value of assets, B/V , the volatility of assets, σA , time to maturity, T − t, and the riskless rate. To estimate B V, we take the ratio of the book value of debt (sum of COMPUSTAT items 9 (long-term debt) and 34 (debt due within a year)) to the quasi-market value of assets (sum of COMPUSTAT items 9 and 34 plus the number of shares outstanding times stock price (both CRSP)). The COMPUSTAT data are taken at the date of the last annual accounting report and the CRSP data are taken on the date of observation. The estimation of asset volatility is a challenging task and here we consider a number of alternatives. First, we compute upper and lower bounds on asset volatility as follows. The maximum equates asset and equity volatility, σE , i.e., assuming zero leverage. The minimum is calculated as σE (1 − L), where L is the market leverage, i.e., assuming that the debt bears no asset risk. In this case, the theoretical hedge ratio is zero. A more realistic estimate of asset volatility recognizes that debt bears some asset risk and that equity and debt covary. For firm j at time t we have: 2 2 2 2 2 σd Ajt = (1 − Ljt ) σEjt + Ljt σDjt + 2Ljt (1 − Ljt )σED,jt , (5) where σDjt is the time t volatility of firm j’s debt, and σED,jt is the time t covariance between returns on firm j’s debt and equity.12 We could estimate firm j’s debt volatility using the returns on each of firm j’s bonds but this approach has two drawbacks. First, it assumes that all of a firm’s outstanding debt has the same volatility as its publicly traded debt. Second, much corporate debt is relatively illiquid and some of the observed volatility may be spurious. 12 This calculation assumes again that leverage is measured instantaneously. 15 We therefore estimate equation (5) as follows. Firm j’s equity volatility at time t is estimated as the time series volatility of returns on firm j’s equity using three years of data up to month t. For the volatility of returns on firm j’s debt we first calculate the average volatility of debt returns by credit rating. Thus, for rating category BBB and firm j, for example, we take the returns on firm j’s debt for each month that the debt was rated BBB at the start of the month. If the bond in question is rated BBB in at least 15 months we then compute the time series volatility. Averaging these volatilities over all firms we obtain the average volatility for BBB debt. The volatility of firm j’s debt in month t is then set equal to the average volatility for the rating category of firm j at month t.13 . The covariance of equity and debt returns, σED,jt , is estimated as ρED,jt σD,jt σE,jt where ρED,jt is estimated in a similar way to σD,jt . Finally, we take time to maturity as equal to the median time to maturity for each rating class and the riskless rate equal 5%. Table VII reports summary statistics for these estimates. As expected, leverage is higher for lower rating categories and, similarly, so is equity volatility. The average values of equity volatility and quasi-market leverage are broadly consistent with similar results reported in other studies. Deleveraging equity volatility using L but taking no account of the asset risk borne by debtholders (the third panel of Table VII) results in estimates of asset volatility that are relatively constant across the rating categories. The fourth panel of Table VII gives estimates of σ d At using equation (5) and the method described above. Here the mean values of asset volatility are quite similar for investment grade bonds (22% to 24%) but noticeably higher for junk bonds: 27% for BB, 29% for B and 30% for C. The range of values, as measured by the 5% and 95% quantiles, is also much wider for the lower rated bonds, e.g.,12-33% for AA vs. 14-56% for B. Table VIII shows estimates of the hedge ratio hE (σE ). The first two panels set σA 13 One can think of this as a form of “switching regime” where the volatility for firm j switches between the volatilities of the different rating categories 16 equal to σE and (1 − L)σE respectively. The final panel shows estimates of the hedge ratio using the estimates of σA derived using equation (5). As expected, these rise monotonically as the rating category declines. IV.4 Testing Merton Model Predictions of Hedge Ratios We now use our estimates of asset volatility to test more formally whether hedge ratios from the Merton model are consistent with the empirical relation between equity and corporate bond returns. For firm j we take the estimate of time t asset volatility described above and use this as an input to the calculation of the time t hedge ratio, hE,jt (c σA ).14 We then estimate the following regression for each firm, j: rj,t = αj,0 + βj,E hE,jt (c σA )rE,t + αj,rf rf 10y,t . (6) Under the null hypothesis that the Merton model correctly estimates the sensitivity of returns on firm j’s debt to firm j’s equity, the coefficient βj,E should be unity. The results are given in Table IX. For the entire sample the mean estimate of βj,E is 1.206. The t-statistic against unity, the value of βj,E under the null is 1.156. For the six rating categories the mean value of βj,E is different from unity in only two cases: BB where the mean value if 2.498 and CCC where the mean is 0.415. For the other four categories the mean value ranges from 0.55 (AA) to 1.54 (B) and none is significantly different from one. These results are supportive of the structural approach, and the Merton model in particular, in a way that previous analyses of the level of prices or credit spreads have not been. They are also complementary to the results recently obtained by Leland (2002) who shows that the default frequency predictions of structural models are also broadly consistent with the data. Apart from the size of the yield spread, there is a further prediction of structural 14 The other inputs – book leverage, time to maturity and the riskless rate – are as described earlier. 17 models that is inconsistent with the results in Table IX. This is the R̄2 in the regression which, if the Black-Scholes conditions supporting the structural approach apply, should be much higher. In our simulations reported in Table VI the R̄2 varied from 0.65% for AAA-A to 0.936% for BB-CCC.15 In Table IX the R̄2 are much lower (11% for CCC and 37% for AA). The low variation in the fraction of rate of return volatility accounted for by equity returns and interest rates has a number of possible explanations. One is simply that it reflects noise in the bond return data (or, possibly in the equity or riskless bond data). This almost certainly accounts for some of the unexplained variability of corporate debt returns. Another is that the model is mis-specified and that either the functional form of the hedge ratio is incorrect or that other variables are necessary to account for the credit exposure of corporate debt. For example, the Merton could hold except that volatility is stochastic and other variables are necessary to predict volatility. Finally, returns on corporate debt could be related to other variables in a way that is not directly related to credit risk. For example, returns on corporate debt might be related to variables that proxy for changes in liquidity. V Other Determinants of Returns on Corporate Bonds In this section we consider the impact of variables that other authors have found to be significant explanators of returns on corporate bond. These include (i) changes in the 10-year minus 2-year yield spread on US Treasuries, (ii) the return on the S&P 500 index, (iii) changes in the VIX index of implied volatility of options on the S&P 100 index and (iv & v) the Fama-French SMB and HML factors. All five factors are included in the recent study by Collin-Dufresne et. al. (2001) and the Fama-French factors are included by Elton et. al. (2001). The results are shown in Table X. The mean coefficients on 10-year Treasury returns are very similar to those in Table IV and those on equity are also similar but little lower. 15 Note that in this procedure interest rate is held constant. 18 The mean coefficients on the S&P are not significant except for the A credit category. The mean coefficients on HML are significant for the whole sample and for three out of the six rating subsamples while those on SMB are significant for the whole sample and all the subsamples except for CCC. Perhaps the most interesting results, however, are those for changes in the VIX volatility index. Collin-Dufresne et. al. (2001) had previously found this variable to be significant in regressions of changes in yield spreads on a very similar list of regressors. The mean coefficients for ∆(V IX) are significant in every case except for CCC; in fact, the t-statistics are higher than those on changes in the firm’s own equity for all the investment grade subsamples. A natural interpretation of the role of ∆(V IX) in the regression is as a proxy for changes in the volatility of equity. In this interpretation VIX is related to a bond’s credit exposure via its effect on the default put. However, when we examine the magnitude of the coefficients for the different rating subcategories we see that they are essentially the same. For example, the mean coefficient for AAA and AA are -0.043 and -0.060 respectively and for BBB and BB are -0.056 and -0.054 respectively. If ∆(V IX) were acting as a proxy for changes in equity volatility the sensitivity for the lower credit categories, as in the case of the coefficients on equity returns, would be much larger.16 We find, therefore, that the change in the VIX index is a variable that, while strongly related to returns returns on credit risky bonds, is apparently unrelated to credit risk. The R̄2 in Table X are significantly higher than in Table IV but particularly so for the non-investment grade bonds: the average R̄2 for BB and B increases to 28% from 20%. The precise rôle of ∆(V IX) and the other variables is, at this point, unclear but it seems highly unlikely that is connected with credit exposure. 16 We have also carried out cross-sectional regressions of the individual coefficients on ∆(V IX) on simple descriptors of credit risk, such as equity volatility and leverage, and found no relation. 19 VI Conlusion This paper studies the ability of structural models to explain excess returns on corporate bonds and the main question we ask is whether these models provide accurate predictions of hedge ratios. Using data on monthly returns for a large sample of U.S. corporate bonds over a five-year period, we find those variables included in structural models – returns on the issuing firm’s equity and on riskless bonds – explain a large fraction of returns on investment grade bonds and a smaller but significant fraction for high yield bonds. Further, and this is the main result of the paper, we find that, for most rating categories, the equity ratios predicted by the Merton model are not rejected by time series data. The next step is to account for other factors. We include in our regression variables that in previous studies have been shown to influence corporate bond prices. The variables we use are: (i) changes in the 10-year minus 2-year yield spread on US Treasuries, (ii) the return on the S&P 500 index, (iii) changes in the VIX index of implied volatility of options on the S&P 100 index and (iv & v) the Fama-French SMB and HML factors. We find that none of the included variables undermines the significance of either the risk-free rate or equity. Our main result here, and the second main result of the paper, is that changes in the VIX index have an impact on corporate bond returns that is both significant and apparently unrelated to a bonds exposure to credit risk. It seems clear, therefore, that returns on credit risky bonds are systematically related to at least one factor that lies outside standard measures of “credit risk”. Whether there are other factors, and the precise role of ∆(V IX) in the determination of risky bond prices, is a question for further research. 20 References [1] Anderson Ronald W., and Suresh M. Sundaresan, 1996, “The Design and Valuation of Debt Contracts”, Review of Financial Studies, 9, 37-68. [2] Blume, M. E., D. B. Keim, and S. Patel, 1991, “Returns and Volatility of Low-Grade Bonds: 1977–1989”, Journal of Finance, 46. [3] Campbell, John Y., and John Ammer, 1993,“What Moves the Stock and Bond Markets? A Variance Decomposition for Long-Term Asset Returns,” Journal of Finance, 48, 3–37. [4] Collin-Dufresne Pierre, and Robert S. Goldstein, 2001, “Do Credit Spreads Reflect Stationary Leverage Ratios?”, Journal of Finance, 56, 1929-1957. [5] Collin-Dufresne Pierre, Robert S. Goldstein, and J.Spencer Martin, 2001, “The Determinants of Credit Spread Changes”, Journal of Finance, 56, 2177-2207. [6] Elton Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, 2001, “Explaining the Rate Spread on Corporate Bonds”, Journal of Finance, 56, 247-277. [7] Eom Young Ho, Jean Helwege, and Jing-zhi Huang, 2003, “Structural Models of Corporate Bond Pricing: An Empirical Analysis”, Review of Financial Studies, forthcoming. [8] Huang, Jing-zhi, and Ming Huang, 2002, “How Much of the Corporate-Treasury Yield Spread is Due to Credit Risk?”, working paper, Pennsylvania State University. [9] Jones P.E., S.P. Mason, and E. Rosenfeld, 1984, “Contigent Claims Analysis of Corporate Capital Structures: An Empirical Analysis”, Journal of Finance, 39, 611-25. [10] Kim, J., Ramaswamy, and S. Sundaresan, 1993, “Does Default Risk in Coupons Affect the Valuation of Corporate Bonds?: A Contingent Claims Model”, Financial Management, 117–131. 21 [11] Leland, Hayne E., 2002, “Predictions of Expected Default Frequencies in Structural Models of Debt”, Venice Conference on Credit Risk, Sept 2002 [12] Longstaff Francis A., and Eduardo S. Schwartz, 1995, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt”, Journal of Finance, 50, 789–819. [13] Mella-Barral Pierre, and William R.M. Perraudin, 1997, “Strategic Debt Service”, Journal of Finance, 52, 531–556. [14] Merton Robert C., 1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”, Journal of Finance, 29, 449-470. 22 Table I Summary statistics for the entire data sample This table reports summary statistics for the entire sample of corporate debt returns over period 12.1996–07.2002. The number of observations is given in thousands. T − t is the time to maturity remaining on the date of each observation. The coupon rate is in %. Volume is in million $US dollars. % in CRSP reports the fraction of observations that are matched with CRSP and COMPUSTAT. Observations Issuers Issues Mean T − t Median T − t Mean Coupon Median Coupon Mean Volume Median Volume % in CRSP All 323.53 2894 9049 10.60 7.04 7.88 7.50 281.52 200 49.96 AAA 7.23 77 301 11.90 7.34 7.37 7.20 338.62 250 13.80 AA 32.20 240 1154 11.13 6.50 7.07 6.88 337.90 200 39.52 23 A 117.17 824 3811 10.94 6.80 7.31 7.13 293.11 200 50.41 BBB 93.38 1035 3332 12.10 7.88 7.58 7.38 269.72 200 55.92 BB 26.93 667 1515 9.36 6.67 8.42 8.36 247.53 200 52.26 B 35.18 1084 1738 6.94 6.97 10.09 10 246.87 179.80 49.97 CCC 11.41 568 879 6.74 6.30 10.69 10.63 250.92 200 43.50 Table II Summary statistics for the final sample This table reports summary statistics for the final sample (the selection procedure is described in the paper) of corporate debt returns over period 12.1996–07.2002. The number of observations is given in thousands. T − t is the time to maturity remaining on the date of each observation. The coupon rate is in %. Volume is in million $US dollars. % in CRSP reports the fraction of observations that are matched with CRSP and COMPUSTAT. Observations Issuers Issues Mean T − t Median T − t Mean Coupon Median Coupon Mean Volume Median Volume % in CRSP All 58.49 493 1362 13.46 8.49 7.76 7.40 259.10 200 100 AAA 0.27 4 6 20.53 24.16 7.26 7.15 321.88 300 100 AA 4.03 30 110 13.87 8.63 6.97 6.75 321.03 250 100 A 18.73 149 560 16.10 12.80 7.41 7.13 266.06 200 100 BBB 23.07 226 687 14.15 8.96 7.49 7.25 258.68 200 100 BB 5.03 121 265 8.71 6.79 8.27 8 247.08 200 100 B 6.20 148 229 7.13 7.13 9.56 9.50 213.59 175 100 CCC 1.16 57 83 8.38 6.88 9.86 9.75 220.90 200 100 Table III Summary statistics on returns This table reports summary statistics for excess and raw returns for the final sample (the selection procedure is described in the paper) of corporate bond returns over period 12.1996– 07.2002. We calculate for each corporate bond j the return between months t and t − 1 as follows: Pj,t + AIj,t + Ij,t Cj /Nj rj,t = , Pj,t−1 where Pj,t is the price of bond j at the end of month t and AIj,t is the accrued interest between t − 1 and t. Since the calculation of the accrued interest restarts with each coupon payment, if the coupon falls between t − 1 and t, Cj /N is added to the price, where Cj is the annual coupon rate and Nj is the coupon frequency per annum of bond j. Ij,t is an indicator function taking the value of 1 if the coupon is due between t − 1 and t. The excess return is then calculated as rj,t = rj,t − rf1m,t , where rf1m,t is the return on one-month Treasury bills. Returns are first calculated for each individual bond and then averaged across bonds. Excess returns are given in parentheses. N is the number of bonds. Mean 5% 95% std N All 0.50 ( 0.11) -0.18 (-0.57) 0.90 ( 0.51) 2.87 ( 2.87) 1362 AAA 0.59 ( 0.20) 0.38 ( 0.00) 0.73 ( 0.34) 1.72 ( 1.70) 6 AA 0.59 ( 0.20) 0.38 (-0.00) 0.75 ( 0.36) 1.85 ( 1.84) 102 A 0.56 ( 0.16) 0.20 (-0.19) 0.76 ( 0.37) 2.15 ( 2.14) 475 24 BBB 0.49 ( 0.10) 0.04 (-0.34) 0.83 ( 0.44) 2.72 ( 2.72) 473 BB 0.48 ( 0.09) -1.40 (-1.80) 0.97 ( 0.59) 3.36 ( 3.36) 124 B 0.34 (-0.05) -2.51 (-2.91) 1.29 ( 0.91) 5.16 ( 5.16) 172 CCC -0.22 (-0.62) -4.07 (-4.49) 3.14 ( 2.73) 9.73 ( 9.74) 10 Table IV Regressions of Excess Returns This table reports regressions of excess returns on corporate bonds over period 12.1996–07.2002. ret rf10y is the excess return on the 10-year constant maturity U.S. Treasury bond. E ret is the excess return on the issuer’s equity and N is the number of observations. Intercept ret rf10y E ret R̄2 N All -0.001 (-4.158) 0.451 (36.655) 0.042 (18.796) 0.375 42.943 (1362) AAA -0.001 (-3.630) 0.748 (6.125) 0.003 (0.685) 0.658 51.500 (6) AA -0.001 (-8.244) 0.690 (28.087) 0.016 (4.635) 0.553 49.039 (102) A -0.001 (-13.142) 0.611 (49.690) 0.021 (10.322) 0.471 43.137 (475) BBB -0.001 (-4.951) 0.507 (27.826) 0.039 (11.038) 0.352 42.655 (473) BB 0.002 (2.876) 0.201 (3.816) 0.074 (6.725) 0.254 41.968 (124) B 0.001 (1.961) -0.066 (-1.845) 0.100 (10.895) 0.153 40.244 (172) CCC -0.006 (-1.073) -0.380 (-1.475) 0.109 (2.794) 0.216 38.500 (10) Table V Sensitivities of bond returns to equity in the Merton (1974) model This table reports the sensitivities of bond returns to equity in the Merton (1974) model for different values of leverage and asset volatility. Leverage is the ratio of the face value of debt to the market value of assets. σA is asset volatility. Leverage and asset volatility are given in %. The riskless rate is assumed to be equal to 5%, time to maturity 10 years. Sensitivities are given in 0.01%. The procedure is described in the paper. t-statistics are in parentheses. σA Leverage 10 20 30 40 50 60 70 10 15 20 25 30 40 50 0 (18.40) 0 ( 4.39) 0 ( 9.66) 0 (10.74) 0.01 ( 7.57) 0.04 (15.08) 0.20 (12.61) 0 ( 2.42) 0 ( 7.93) 0.02 (12.67) 0.15 (10.28) 0.55 (15.88) 1.33 (17.82) 2.41 (23.32) 0 ( 7.90) 0.07 (13.12) 0.42 (18.21) 1.35 (20.91) 2.96 (23.96) 4.44 (27.88) 6.33 (34.45) 0.05 ( 9.88) 0.66 (13.67) 2.35 (17.17) 4.31 (26.88) 6.23 (32.97) 8.63 (35.39) 10.31 (41.62) 0.37 (16.33) 2.44 (18.82) 5.19 (25.23) 7.72 (33) 10.31 (39.25) 12.70 (42.87) 14.75 (46.37) 3.66 (20.55) 7.91 (27.80) 12.02 (32.86) 15.57 (40.93) 18.16 (48.93) 20.36 (54.62) 21.53 (56.57) 8.70 (27.54) 14.18 (38.65) 20.10 (50.08) 22.64 (53.12) 24.83 (58.37) 26.64 (61.85) 28.72 (74.88) 25 Table VI Sensitivities of debt returns to equity implied by the Merton (1974) model for the actual data set This table reports sensitivities of bond returns to equity in the Merton (1974) model. The sensitivities were obtained using a simulation, as described in the paper. Sensitivities are given in 0.01%. t-statistics are in parentheses. hE R̄2 All 0.046 (29.50) 0.74 AAA-A 0.011 (17.95) 0.65 BBB 0.040 (22.87) 0.78 BB-CCC 0.153 (32.54) 0.93 Table VII Leverage and Volatilities This table reports results of the first step of a more extensive analysis of hedge ratios. σE is the historical equity volatility. L is market leverage. σc A is the estimated asset volatility, hE is the hedge ratio calculated for three different estimates of asset volatility. Volatilities and leverage are given in % and hedge ratios are given in 0.01%. N is the number of observations. The details of the estimation procedure are given in the paper. L Mean Median Std. Dev. 5% quantile 95% quantile σE Mean Median Std. Dev. 5% quantile 95% quantile (1 − L)σE Mean Median Std. Dev. 5% quantile 95% quantile σcA Mean Median Std. Dev. 5% quantile 95% quantile N All AAA AA A BBB BB B CCC 0.346 0.301 0.232 0.035 0.804 0.047 0.026 0.143 0.014 0.042 0.162 0.110 0.169 0.028 0.458 0.239 0.208 0.165 0.036 0.564 0.364 0.342 0.204 0.057 0.754 0.439 0.449 0.222 0.064 0.802 0.587 0.614 0.219 0.192 0.886 0.751 0.817 0.209 0.343 0.975 0.361 0.326 0.165 0.186 0.669 0.251 0.241 0.045 0.188 0.346 0.268 0.258 0.074 0.156 0.402 0.291 0.277 0.092 0.177 0.468 0.348 0.328 0.117 0.193 0.585 0.438 0.421 0.142 0.240 0.716 0.543 0.517 0.193 0.285 0.905 0.803 0.753 0.376 0.504 1.205 0.221 0.204 0.110 0.080 0.420 0.240 0.236 0.056 0.183 0.332 0.222 0.222 0.071 0.111 0.328 0.219 0.207 0.081 0.117 0.365 0.219 0.200 0.110 0.086 0.448 0.246 0.217 0.138 0.077 0.524 0.222 0.189 0.148 0.060 0.524 0.194 0.135 0.193 0.020 0.549 0.239 0.219 0.106 0.118 0.440 58488 0.240 0.237 0.055 0.184 0.333 272 0.224 0.223 0.069 0.116 0.329 4027 0.223 0.211 0.080 0.125 0.366 18729 0.230 0.211 0.106 0.108 0.453 23068 0.270 0.244 0.128 0.117 0.530 5031 0.287 0.258 0.133 0.136 0.557 6203 0.300 0.262 0.184 0.095 0.622 1158 26 Table VIII Hedge Ratios This table reports results of the first step of a more extensive analysis of hedge ratios. σE is the historical equity volatility. L is market leverage. σc A is the estimated asset volatility, hE is the hedge ratio calculated for three different estimates of asset volatility. Volatilities and leverage are given in % and hedge ratios are given in 0.01%. N is the number of observations. The details of the estimation procedure are given in the paper. hE (σE ) Mean Median Std. Dev. 5% quantile 95% quantile hE ((1 − L)σE ) Mean Median Std. Dev. 5% quantile 95% quantile hE (σcA ) Mean Median Std. Dev. 5% quantile 95% quantile N All AAA AA A BBB BB B CCC 0.095 0.053 0.105 0 0.306 0.006 0 0.022 0 0.029 0.023 0.001 0.055 0 0.129 0.047 0.011 0.071 0 0.213 0.092 0.064 0.089 0 0.265 0.138 0.133 0.098 0.001 0.312 0.221 0.230 0.097 0.045 0.367 0.334 0.343 0.074 0.208 0.438 0.020 0.003 0.037 0 0.095 0.002 0 0.006 0 0.017 0.005 0 0.014 0 0.034 0.012 0.001 0.026 0 0.066 0.018 0.005 0.032 0 0.083 0.030 0.013 0.042 0 0.118 0.045 0.023 0.056 0 0.163 0.053 0.021 0.073 0 0.196 0.028 0.007 0.044 0 0.121 58488 0.002 0 0.006 0 0.017 272 0.006 0 0.014 0 0.036 4027 0.013 0.001 0.027 0 0.070 18729 0.021 0.008 0.033 0 0.091 23068 0.040 0.027 0.045 0 0.135 5031 0.083 0.071 0.055 0.011 0.191 6203 0.123 0.116 0.073 0.016 0.250 1158 Table IX Hedge ratio regressions This table reports results of a regression analysis of hedge ratios. The regression is rj,t = αj,0 + βj,E hE,j,t rE,t + αj,rf rf 10y,t , where hE,j,t is the hedge ratio for firm j at time t as implied by the model. Under the null ret is the excess hypothesis that the Merton model holds, βj,E is equal to one (100 bp). rf10y return on the 10-year constant maturity U.S. Treasury bond. The details of the estimation procedure are given in the paper. N is the number of observations. Intercept βE ret rf10y R̄2 N All -0.001 (-4.554) 1.206 (1.156) 0.369 (18.180) 0.266 42.076 (686) AAA (—) (—) (—) (—) (—) (—) (—) 35 (1) AA -0.002 (-6.764) 0.552 (-1.304) 0.815 (25.734) 0.372 48.536 (28) 27 A -0.002 (-8.601) 1.173 (0.608) 0.688 (34.809) 0.371 42.902 (143) BBB -0.001 (-4.978) 0.787 (-0.629) 0.479 (22.968) 0.289 41.569 (288) BB 0.001 (1.312) 2.498 (2.968) 0.140 (2.172) 0.203 42.976 (83) B -0.001 (-1.207) 1.540 (1.762) -0.116 (-2.165) 0.128 40.471 (136) CCC -0.007 (-0.854) 0.415 (-2.159) -0.408 (-1.386) 0.108 41.714 (7) Table X Regression of Excess Returns This table reports results of regression analysis of excess returns on corporate bonds over period ret 12.1996-07.2002. rf10y is the excess return on the 10-year constant maturity U.S. Treasury ret bond. E is the excess return on the issuer’s equity. ∆(rf10y − rf2y ) is the change in the slope of the term structure (the difference between the yield on ten-year and two-year constantmaturity U.S. Treasury bonds). S&P ret is the return on the S&P index. VIX is the implied option volatility on the S&P100 index options. SMB and HML are the Fama-French Small minus Big and High minus Low factors. N is the number of observations. t-statistics are given in parentheses. Int ret rf10y ∆(Spd) E ret S&P ret ∆(V ix) SMB HML R̄2 N All -0.001 (-8.628) 0.509 (41.557) -0.015 (-11.239) 0.030 (13.196) 0.008 (1.691) -0.053 (-13.073) 0.104 (19.429) 0.023 (5.558) 0.463 42.943 (1362) AAA -0.001 (-4.049) 0.796 (6.137) -0.006 (-0.958) 0.002 (0.426) 0.023 (1.487) -0.043 (-4.414) 0.052 (2.518) 0.021 (1.532) 0.700 51.500 (6) AA -0.002 (-10.237) 0.743 (27.578) -0.008 (-3.419) 0.009 (2.423) -0.002 (-0.253) -0.060 (-9.083) 0.077 (8.165) 0.017 (4.154) 0.631 49.039 (102) 28 A -0.002 (-17) 0.659 (53.560) -0.012 (-7.504) 0.008 (4.249) 0.019 (4.157) -0.045 (-10.712) 0.092 (15.424) 0.012 (3.241) 0.553 43.137 (475) BBB -0.002 (-7.561) 0.573 (31.885) -0.009 (-4.369) 0.028 (7.377) -0.002 (-0.178) -0.056 (-8.282) 0.104 (13.536) 0.008 (1.145) 0.455 42.655 (473) BB 0 (1.077) 0.291 (6.223) -0.027 (-6.261) 0.054 (5.304) 0.033 (1.913) -0.054 (-3.314) 0.182 (6.929) 0.080 (5.400) 0.347 41.968 (124) B 0.001 (0.905) -0.018 (-0.468) -0.027 (-4.526) 0.091 (9.198) -0.014 (-0.593) -0.059 (-3.071) 0.100 (3.986) 0.040 (1.923) 0.221 40.244 (172) CCC -0.004 (-0.931) -0.416 (-1.805) -0.131 (-3.941) 0.110 (2.502) 0.146 (0.892) -0.095 (-1.035) 0.124 (1.020) 0.303 (4.098) 0.247 38.500 (10)