Asset volatility

Our analysis is based on a comprehensive panel of U.S. corporate bond data, which includes all the constituents of (i) Barclays U.S. Corporate Investment Grade Index and (ii) Barclays U.S. High Yield Index. The data includes monthly returns and bond characteristics from September 1988 to February 2013. We exclude financial firms with SIC codes between 6000 and 6999.

Given that corporate issuers often issue multiple bonds and that our analysis is directed at measuring asset volatility of the issuer, we need to select a representative bond for each issuer. To do this, we follow the criteria of Haesen et al. (2013). We repeat this exercise every month for our sample period. The criteria used for identifying the representative bond are selected so as to create a sample of liquid and cross-sectionally comparable bonds. Specifically, we select representative bonds on the basis of (i) seniority, (ii) maturity, (iii) age, and (iv) size.

First, we filter bonds by seniority. Because most companies issue the majority of their bonds as senior debt, we select only bonds corresponding to the largest rating of the issuer. To do this, we first compute the amount of bonds outstanding for each rating category for a given issuer. We then keep only those bonds that belong to the rating category that contains the largest fraction of debt outstanding. These bond tends to have the same rating as the issuer. Second, we then filter based on maturity. If the issuer has bonds with time to maturity between 5 and 15 years, we remove all other bonds for that issuer from the sample. If not, we keep all bonds in the sample. Third, we then filter based on time since issuance. If the issuer has any bonds that are at most two years old, we remove all other bonds for that issuer. If not, we keep all bonds from that issuer in the sample. Finally, we filter based on size. Of the remaining bonds, we pick the one with the largest amount outstanding.²

Our resulting sample includes 121,300 unique bond-month observations, corresponding to 5362 bonds issued by 1504 unique firms. Table 1 Panel A shows the industry composition of the sample, using Barclays Capital’s industry definitions. Approximately 35% of the sample firms are consumer products firms. Capital goods firms and basic industry make up another 20% of the sample. Sample bonds have an average option-adjusted spread (OAS) of 3.31% over the sample period and an average option adjusted duration of 5.16 years (Table 1, Panel B). Appendix I defines these variables as well as other variables used in the paper in more detail.

We estimate the probability of bankruptcy based on a large sample of Chapter 7 and Chapter 11 bankruptcies filed between 1980 and the end of 2012. We combine bankruptcy data from four main sources: Beaver et al. (2012)⁴; the New Generation Research bankruptcy database (bankruptcydata.​com); Mergent FISD; and the UCLA-Lo Pucki bankruptcy database.

We use a discrete time-hazard model and include three types of observations in the estimation: nonbankrupt firms, years before bankruptcy for bankrupt firms, and bankruptcy years (Shumway 2001). Our dependent variable equals 1 if a firm files for bankruptcy within one year of the end of the month and zero otherwise. We keep the first bankruptcy filing and remove from the sample all months after this filing.

Following Correia et al. (2012), we use quarterly financial data to compute the default barrier and update market data on a monthly basis to obtain monthly estimates of the probabilities of bankruptcy. Market variables are measured at the end of each month, and accounting variables are based on the most recent quarterly information reported before the end of the month. We winsorize all independent variables at 1% and 99%. We ensure that all independent variables are observable before the declaration of bankruptcy. Furthermore, to ensure that prediction is made out of sample and to avoid a potential bias of ex post over-fitting the data, we estimate coefficients using an expanding window approach. We convert the different scores into probabilities as follows: Prob = e^score/1 + e^score. All of the models are nonlinear transformations of various fundamental and market data.

The primary regression model for estimating bankruptcy over the next 12 months is as follows: \( \ln \left(\frac{V_{it}}{X_{it}}\right) \) is a measure of dollar distance to default barrier (akin to an inverse measure of leverage). We compute V _it as the sum of the market value of the firm’s equity and the book value of debt. We compute our default barrier, X _it , as the sum of short-term debt (DLCQ) and half of long-term debt (DLTTQ) as reported at the most recent fiscal quarter (e.g., Bharath and Shumway 2008). Exret _it is the excess equity return over the value-weighted market return over the previous 12 months. ln(E _it ) is the logarithm of the market value of equity measured at the start of the forecasting month. P _5, it is an estimate of the 5th percentile of the distribution of RNOA. It is calculated as described in Appendix III, using the quantile regressions employed by Konstantinidi and Pope (2016). P _5, it is a measure of left-tail risk in profitability. Skew _it is an estimate of the skewness of the distribution of RNOA. Following Konstantinidi and Pope (2016), we estimate skewness as \( \frac{\left({P}_{75}-{P}_{50}\right)-\left({P}_{50}-{P}_{25}\right)}{IQR} \), where IQR is the interquartile range (P ₇₅ − P ₂₅). Accordingly, Skew _it ranges between −1 and 1 and is zero when the distribution of RNOA is symmetric within the interquartile range. Kurt _it is an estimate of the kurtosis of the distribution of RNOA, estimated following Konstantinidi and Pope (2016) as \( \frac{\left({P}_{87.5}-{P}_{62.5}\right)+\left({P}_{37.5}-{P}_{12.5}\right)}{IQR} \). σ _k, it is the respective measure of asset volatility as defined in section 2.3. The choice of independent variables is based on the Merton model of credit spreads to which we add a measure of left-tail risk. We estimate equation (4) using various combinations of our measures of asset volatility over different samples to assess the relative importance of market-based and fundamental-based measures of asset volatility in the context of forecasting bankruptcy.

Our priors for equation (4) are as follows. (i) \( \ln \left(\frac{{\mathrm{V}}_{\mathrm{it}}}{{\mathrm{X}}_{\mathrm{it}}}\right) \) is expected to be negatively associated with bankruptcy likelihood (the further the market value of assets is from the default barrier the lower the likelihood of hitting that barrier in the next 12 months). (ii) Exret_it is expected to be negatively associated with bankruptcy likelihood (assuming there is information content in security prices, decreases in security prices should be associated with increased bankruptcy likelihood). (iii) ln(E_it) is expected to be negatively associated with bankruptcy likelihood (large firms offer better diversification and better realizations of asset values in the event of default). (iv) P _5, it is expected to be negatively associated with bankruptcy likelihood (the higher the 5th percentile of the RNOA distribution, the lower the probability that asset value will fall below the book value of debt). (v) Skew _it is expected to be negatively associated with bankruptcy likelihood (the more negatively skewed the distribution of earnings, the higher the likelihood the asset value will fall below the book value of debt). (vi) Kurt _it is expected to be positively associated with bankruptcy likelihood (higher kurtosis indicates that the density of the tails of the distribution is higher than what would be expected under a normal distribution). (vii) σ_k, it is expected to be positively associated with bankruptcy likelihood (the greater the volatility of the asset value the greater the chance of passing through the default barrier).

In an alternative specification, we also control for the level of option-adjusted spreads (OAS _it ) as a market based measure of credit risk. To the extent that credit market participants incorporate fundamental volatility in assessing credit risk, OAS _it could subsume the fundamental volatility measures.

Given that a measure of asset volatility is useful in forecasting bankruptcy and under the assumption that security prices in the secondary credit market are reasonably efficient, we also test how different combinations of measures of asset volatility can explain cross-sectional variation in credit spreads. We view the analysis of credit spreads as supporting evidence for assessing the information content of fundamental- and market-based measures of asset volatility.

We do this via two approaches. First, we estimate an unconstrained cross-sectional regression where we include multiple measures of determinants of credit spreads in a linear model. Second, we estimate a constrained cross-sectional regression where we combine our various measures of asset volatility into measures of distance to default, which are in turn mapped to an implied credit spread following the approach of Crouhy et al. (2000); Kealhofer (2003); and Arora et al. (2005). A benefit of the constrained approach is that it combines the dollar distance to default, \( \ln \left(\frac{{\mathrm{V}}_{\mathrm{it}}}{{\mathrm{X}}_{\mathrm{it}}}\right) \), with measures of asset volatility, σ_k, it, to better identify closeness to the default threshold. An unconstrained regression cannot capture the inherent nonlinear relations between leverage, asset volatility, defaults (bankruptcy), and credit spreads.

For the unconstrained approach, we estimate the following regression model.

OAS_it is the option-adjusted spread for the respective bond as reported in the Barclays Index. An intercept is not reported as we include time fixed effects. In addition to the determinants of bankruptcy, i.e., \( \ln \left(\frac{{\mathrm{V}}_{\mathrm{it}}}{{\mathrm{X}}_{\mathrm{it}}}\right) \), Exret_it, ln(E_it), P_5, it , Skew _it , Kurt _it , and σ_k, it, which are all issuer-level determinants of credit risk, we also include issue-specific determinants of credit risk and liquidity that will influence the level of credit spreads. Specifically, our additional controls include (i) Rating_it, the issue-specific rating (higher rated issues are expected to have higher credit spreads, given that we code ratings to be increasing in risk), (ii) Age_it, the time since issuance in years (liquidity is decreasing for progressively off-the-run securities, so we expect credit spreads to be increasing in time since issuance), and (iii) Duration_it, the option-adjusted duration of the issue (for the vast majority of corporate issuers the credit term structure is upward sloping so we expect credit spreads to increase with duration; see Helwege and Turner 1999).

For the constrained approach, we then estimate the following regression model.

\( {\mathrm{CS}}_{\upsigma_{\mathrm{k},\mathrm{it}}} \) is the theoretical credit spread for the k^th measure of asset volatility. The estimation of theoretical credit spreads entails six main steps (which are described in detail in Appendix II). (1) We standardize each asset volatility measure and match its moments to the moments of weighted historical asset volatility, \( {\sigma}_A^{\omega } \). (2) We construct estimates of distance to default, based on each asset volatility measure. (3) We empirically map each distance-to-default measure to our bankruptcy data, using a discrete time hazard model to generate a forecast of physical bankruptcy probability (see equation (A.1) of Appendix II).⁵ (4) We compute a cumulative physical bankruptcy probability by cumulating default probabilities over the duration of the bond. (5) We convert each cumulative physical probability measure into a risk-neutral measure, by adding a risk-premium (see equation (A.2) of Appendix II). (6) Based on this risk-neutral measure and the expected recovery rate (which is assumed to be constant), we calculate theoretical credit spread as in equation (A.3).

We obtain a different theoretical credit spread for each asset volatility measure. We estimate two additional credit spreads, \( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \)and \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \), based on the combination of our seven fundamental volatility measures (i.e., σ_F, σ_IQR, σ_FEPS, σ_MARGIN, σ_TURNOVER, σ_OI GROWTH, σ_{SALES GROWTH}). \( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \) and \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \) differ in the way in which the different volatilities are combined. To obtain \( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \), we take the average of the seven fundamental volatility measures after step (1) above (i.e., after matching their respective moments to \( {\upsigma}_{\mathrm{A}}^{\omega } \)). We then follow steps (2) to (6), using this average as a measure of fundamental volatility. In contrast, to calculate \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \), we first follow steps (1) to (3) to obtain estimates of the physical default probabilities corresponding to each of the seven fundamental volatility measures. We then take the average of these physical default probabilities and follow steps (4) to (6) based on this average. The average monthly correlation between \( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \) and \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \) is above 0.9 (Table 1 Panel E).

\( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \) and \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \) exhibit an average Pearson (Spearman) correlation with market-based credit spreads (\( {\mathrm{CS}}_{\upsigma_{\mathrm{A}}^{\upomega}},{\mathrm{CS}}_{\upsigma_{\mathrm{A}\mathrm{I}}^{\upomega}} \)) of 0.7740 (0.5938) and an average Pearson (Spearman) correlation of 0.7172 (0.6165) with observed credit spreads. The high correlation with OAS suggests that our structured use of leverage and asset volatility as outlined in Appendix II is an effective way to aggregate market and fundamental information for credit valuation purposes.

Theoretical spreads based on historical security data or option-implied volatility exhibit a higher correlation with observed spreads than theoretical spreads based on fundamental accounting data. In particular, OAS exhibits an average Pearson (Spearman) correlation with market-based spreads (\( {\mathrm{CS}}_{\upsigma_{\mathrm{A}}^{\upomega}},{\mathrm{CS}}_{\upsigma_{\mathrm{A}\mathrm{I}}^{\upomega}} \)) of 0.7664 (0.6946) and an average Pearson (Spearman) correlation with accounting-based spreads (\( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \), \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \)) of 0.7172 (0.6165). Also note that \( {\mathrm{CS}}_{\upsigma_{\mathrm{AVG}}} \) and \( {\mathrm{CS}}_{{\mathrm{PROB}}_{\mathrm{AVG}}} \)exhibit stronger correlations with OAS than \( {\mathrm{CS}}_{\upsigma_{\mathrm{F}}} \). (The Pearson (Spearman) correlation between \( {\mathrm{CS}}_{\upsigma_{\mathrm{F}}} \) and OAS is 0.6288 (0.5781)). This suggests that there is value to conducting a deeper financial statement analysis and combining different fundamental volatility measures.

Table 2 reports the estimation results of regression equation (4). The sample size used for the basis of estimating equation (4) is 81,802 bond-month observations (in specifications with σ_I the sample is reduced to 61,132 observations, as σ_I is only available from 1996 onward). The sample is further reduced in specifications that include σ_FEPS and hence require availability of IBES data.

Across all specifications, we find expected relations for our primary determinants: bankruptcy likelihood is decreasing in (i) distance to default barrier, \( \ln \left(\frac{{\mathrm{V}}_{\mathrm{it}}}{{\mathrm{X}}_{\mathrm{it}}}\right) \), (ii) recent equity returns, Exret_it, and (iii) firm size, ln(E_it). The coefficients on P_5, it, Skew _it , and Kurt _it are insignificant across most specifications. To assess the relative importance of our different measures of asset volatility, we first examine each measure individually after controlling for the same-issuer-level determinants of bankruptcy. Across models (1) to (6) in Table 2, we find that all of the measures of asset volatility are significantly positively associated with the probability of bankruptcy.

To provide a sense of the relative economic significance across the different measures of asset volatility, we report in Panel B of Table 2 the marginal effects for each explanatory variable. Specifically, we hold each explanatory variable at its average value and report the change in probability of bankruptcy for a one standard deviation change in the respective explanatory variable, relative to the full sample unconditional probability of bankruptcy. For example, column (1) in Panel B of Table 2 states that the marginal effect of σ_E is 0.0110. This means that a one standard deviation change in σ_E is associated with a 1.1% increase in bankruptcy probability, relative to the full sample unconditional probability of bankruptcy (0.80%). Comparing marginal effects across explanatory variables reveals that the distance to default barrier is the most economically important explanatory variable. Individually, the most important measure of asset volatility is σ_I (marginal effect of 0.0531 is the largest in the first 6 columns of Panel B of Table 2).

Models (7) to (14) in Panel A combine different measures of asset volatility. We do not include σ_E and σ_I in the same specification due to multi-collinearity (Panel D of Table 1 shows that σ_E and σ_I have a parametric correlation of 0.8814). In model (7), we start with issuer-level determinants (\( \ln \left(\frac{{\mathrm{V}}_{\mathrm{it}}}{{\mathrm{X}}_{\mathrm{it}}}\right) \), Exret_it, ln(E_it), P _5, it , Skew _it , and Kurt _it ) and σ_E. We then add a measure of volatility from the credit markets, σ_D. Combining market-based measures of asset volatility from the equity and credit markets is superior to examining equity market information alone. (The pseudo-R² marginally increases from 39.22% in model (1) to 39.84% in model (7) and from 28.52% in model (2) to 28.53% in model (11).) However, the coefficient on σ_D is not statistically significant when σ_D is combined with σ _I in model (11). In model (8), when we add our first measure of fundamental volatility, σ_F, we find that both σ_D and σ_F are significantly associated with bankruptcy but σ_E is not. When σ_F is added to σ_I and σ_D in model (12), σ_I and σ_F are significant but σ_D is not. Using the interquartile range of the RNOA distribution and the dispersion of analyst forecasts as measures of fundamental volatility in models (9) and (13) and in models (10) and (14), respectively, we find similar results: combining measures of volatility from market and fundamental sources improves explanatory power of bankruptcy prediction models. While we do not run a horse-race between the fundamental volatility measures, we believe this could be an interesting avenue for future research. In an untabulated robustness analysis, we document further that our fundamental volatility measures also improve upon the explanatory power of a bankruptcy prediction model that includes Merton-based volatility and leverage measures (e.g., Bharath and Shumway 2008). This approach takes equity prices, equity volatility, and current leverage as given and then solves iteratively for asset value and asset volatility that price equity as a call option on the asset value of the firm.

In Panel C, we add a control for the spread level, OAS. We find that OAS subsumes σ_E and σ_D, which both cease to be significant in models (1) and (3). Fundamental volatility measures remain significant, both when included by themselves (models (4) to (6)) and when combined with σ_E, σ_I and σ_D (models (8) to (10) and (12) to (14)). Interestingly, the marginal effects reported in panel D of Table 2 reveal that, after controlling for credit spreads, the difference in the relative importance of market-based and fundamental-based measures is more muted. For example, in models (12) to (14) σ_F, σ_IQR and σ_FEPS have similar importance to the market-based measures. The fact that σ_F, σ_IQR and σ_FEPS remain significant, after controlling for OAS could be consistent with the market not paying enough attention to fundamental measures of asset volatility.

In Table 3, we start with a model that includes σ_I, σ_D, and σ_F, in column (1).⁶ We then replace σ_F by the volatility of operating profit margins, σ_MARGIN, and the volatility of asset turnover, σ_TURNOVER, in the spirit of the Dupont profitability decomposition. The Pearson (Spearman) correlation between σ_MARGIN and σ_TURNOVER volatility is 0.0087 (−0.1665) (Table 1 Panel C). When we include both σ_MARGIN and σ_TURNOVER in equation (4), we find that σ_TURNOVER is marginally significant but σ_MARGIN is not (column (2)). We obtain similar results when we control for OAS in column (5)). The volatility of operating income growth σ_OI GROWTH is significant in both specifications (columns (3) and (6)). In unreported analysis, we find similar results with the volatility of sales growth, σ_{SALES GROWTH}, but due to its high correlation with σ_OI GROWTH we do not report these results separately.

One limitation with the traditional discrete hazard model analysis is that it cannot capture nonlinearities and interactions that are likely among the independent variables. As an alternative methodological approach, we analyze our default data using the classification and regression trees methodology developed by Breiman et al. (1984).⁷ Frydman et al. (1985) apply this technique to the prediction of financial distress and document that it outperforms discriminant analysis in out-of-sample tests. The data is recursively split into more homogeneous subsets, using the Gini rule to choose the optimal split at each node of the tree. Based on this approach, we generate a maximal tree and a set of sub-trees. We then use tenfold cross validation to estimate the area under the receiver operating characteristic curve (i.e., AUC) for the different sub-trees and retain the minimal cost tree. The resulting tree structure allows for nonlinear and interactive associations between probability of default and the different explanatory variables, alleviating the concern that documented results are simply due to method variance.

To focus on the relative importance of accounting- and market-based measures of asset volatility, we first apply this technique to a basic set of bankruptcy determinants, i.e., \( \ln \left(\frac{{\mathrm{V}}_{\mathrm{it}}}{{\mathrm{X}}_{\mathrm{it}}}\right) \), Exret_it, ln(E_it), P_5, it , Skew _it , Kurt _it and a representative market-based measure of asset volatility that combines information from implied equity option data and debt market volatility, \( {\upsigma}_{\mathrm{AI}}^{\upomega} \). The CART estimation does not pose the same multicollinearity issues as the discrete hazard model estimation reported in Tables 2 and 3, and therefore we can include all asset volatility measures simultaneously in the model. We thus augment the set of bankruptcy predictors with our seven fundamental volatility measures: σ_F, σ_IQR, σ_FEPS, σ_MARGIN, σ_TURNOVER, σ_OI GROWTH, σ_{SALES GROWTH}.

Panel A of Table 4 reports summary statistics for the predictive ability of the resulting trees. Column (1) serves as the benchmark case where no fundamental-based measures of asset volatility are included. Comparing columns (1) and (2), it is clear that the test-sample (out-of-sample) AUC improves with the inclusion of fundamental-based measures of asset volatility. Note that the test-sample AUC for the augmented model is 0.9337, while the test-sample AUC for the basic model that only includes market volatility is 0.9215. We use bootstrap resampling to test the statistical significance of improvement in AUC. In particular, we construct 100 bootstrap samples and apply CART to each of these samples, thus building 100 different trees for each set of variables. We then compute the difference between the AUC of each of the augmented models and the AUC of the basic model. The 5th percentile of this difference is positive for the augmented model (column (2)), indicating that the improvement in the AUC achieved by incorporating the fundamental volatility measures is statistically significant at conventional levels. The relative cost (the simple sum of type I and type II classification errors) is also reduced by the inclusion of fundamental asset volatility measures. In the base model the relative cost is 0.1716. However, the inclusion of accounting based measures of asset volatility lowers the relative cost measure to 0.1374. The inclusion of OAS in the model (column (3)) does not significantly increase the AUC, with respect to the model that includes fundamental volatility information, and, in contrast, increases the relative cost.

To further understand the economic significance of fundamental-based measures of asset volatility, we compute importance scores for each of the variables in the model (Panel B of Table 4). These scores attempt to measure how much work a variable does in a particular tree. They are calculated as the sum of the improvement that can be attributed to that variable at each node of the tree, weighted by the number of observations passing through that node (i.e., splits lower in the tree with only a smaller fraction of data passing through receive lower scores). For example, suppose that there are N observations in a given tree node (the parent node, t) and that variable s is chosen to split those N observations into two child nodes (t _L  and t _R ). Variable s, together with all the other variables used to recursively split the sample data in the tree, is called a primary splitter. The improvement attributed to variable s in that specific node t is simply ∆R(s, t) = R(t) − R(t _L ) − R(t _H ), with \( R(t)=\frac{1}{N}\sum \limits_{x_n\in t}{\left({y}_n-\overline{y}(t)\right)}^2 \), and effectively reflects a change in the sum of square errors as a result of the split. To compute the variable importance score for variable s, we thus (1) identify all the nodes t in which variable s is used as a splitter, (2) compute the split improvement (∆R(s, t)) for all of these nodes, (3) adjust the split improvement to take into account the percentage of the sample flowing through each node, (4) add all the resulting improvement scores to compute the raw variable importance of variable s, and (5) rank and scale all raw variable importance scores, such that the variable with highest importance receives a score of 100. Following Breiman et al. (1984), we also examine the role that each variable plays as a surrogate. A surrogate is simply a substitute for a primary splitter at a certain node. The surrogate divides the data in a similar way to the primary splitter and may thus be used to replace the primary splitter when the primary splitter is missing. Our total variable importance score considers the role of each variable both as a primary splitter and as a surrogate. It is estimated following the approach described above, except that we now identify all the nodes where CART selects the variable either as a primary splitter or a surrogate and add all the corresponding improvement scores.

Leverage is the most important variable in models (1) and (3). Furthermore, the importance scores of fundamental-based measures of asset volatility are higher than those of market-based volatility measures, both considering just the role of each variable as primary splitter and its combined role as primary splitter and surrogate. When OAS is included in the model (model (3)) it becomes the second highest importance variable (after leverage). While OAS is assigned a total variable importance score of 94.35 in model (3), it has no importance as a primary splitter. This is in contrast with leverage, which has a total variable importance of 100 and a variable importance as a primary splitter of 100. This suggests that, while OAS is not directly used in the prediction tree, it plays an important role as a surrogate, i.e., it could replace leverage and other predictors if they were missing. Most importantly, the variable importance of the fundamental volatility measures remains high when OAS is added to the model, ranging from 7.63 to 43.90, compared to the 2.34 variable importance of \( {\upsigma}_{\mathrm{AI}}^{\upomega} \).

Variable importance scores capture the role played by a variable in a specific tree, and CART trees may be sensitive to the training data. This issue is partially addressed by the fact that we use cross-validation to build test samples and choose the optimal tree. To further circumvent this potential issue and assess the stability of our variable importance scores, we build 100 bootstrap samples and compute variable importance scores for each of these samples. Figure 1 plots the distribution (specifically, the minimum, 25th percentile, median, 75th percentile and maximum) of these scores. Note that the ranking of the variables in each of the panels of Fig. 1 may not exactly correspond to the ranking of the variables in Table 4, Panel B. This is because the ranking in Fig. 1 is based on the median importance of the variable in the 100 trees built using the bootstrap samples, whereas the importance scores reported in Table 4, Panel B are based on the tree built using our original data. Both Figure 1 and Table 4 highlight the importance of fundamental asset volatility for predicting defaults out of sample, when compared to both \( {\upsigma}_{\mathrm{AI}}^{\upomega} \) and the basic set of bankruptcy determinants.