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---

### Key empirical findings
- Firm leverage (Net Debt/Asset) declined by 5.3 percentage points due to Covid-related lockdowns.
- Pre-crisis mean leverage in the sample is 19.6 percent.
- Mean Gross Debt/Asset in the pre-COVID sample is 33.43 percent.
- Debt maturity increased moderately following the onset of COVID-19.
- De-leveraging and rise in maturity are stronger among firms exposed to high financial (rollover) risk.
- Firms most vulnerable to social distancing did not reduce leverage relative to less-vulnerable firms, although they shortened their debt maturity structure.
- Median leverage ratio rose from 10 percent in early 2007 to 21 percent in 2019:Q4, followed by a decline beginning 2020:Q1.
- U.S. GDP dropped by 9.5 percent quarter-over-quarter (noted in context of the shock).

### Heterogeneity and risk channels
- Financial risk (rollover risk) measured via the current portion of long-term debt: firms with higher refinancing needs reduced leverage by approximately 4 percentage points more than less-exposed firms.
- Business risk (exposure to social distancing) measured by growth in sales in 2020:Q2 (bottom quartile classified as exposed): these firms did not exhibit additional de-leveraging, and some became over-leveraged relative to model-implied optimal leverage.
- The de-leveraging effect is robust to alternate measures of leverage (Gross Debt/Asset) and liquidity variations.
- Firms exposed to financial risk compose 6.09 percent of total assets; firms exposed to business risk compose 10.48 percent of total assets.

### Empirical strategy and benchmark results
- Dependent variables: (i) Net Debt/Asset, (ii) Net Long-term Debt/Asset, (iii) Long-term Debt/Total Debt.
- Controls (lagged by one quarter): log(Assets), Tangibility, Market/Book value of Equity, EBIT/Asset.
- Fixed effects: firm 훼i, year 훿y, sector-time 휆jt. Variable Post t = 1 for 2020:Q2-2020:Q4.
- Baseline sample means (Panel A pre-COVID 2018:Q1-2020:Q1):
  - Full Sample: Gross Debt/Asset (%) 33.43, Net Debt/Asset (%) 19.56, Maturity 0.74
  - Firms Exposed to Financial Risk: Gross Debt/Asset (%) 29.28, Net Debt/Asset (%) 2.92, Maturity 0.52
  - Firms Exposed to Business Risk: Gross Debt/Asset (%) 38.65, Net Debt/Asset (%) 23.32, Maturity 0.77
- Post-COVID means (Panel B 2020:Q2-2020:Q4):
  - Full Sample: Gross Debt/Asset (%) 33.91, Net Debt/Asset (%) 16.99, Maturity 0.77
  - Firms Exposed to Financial Risk: Gross Debt/Asset (%) 26.11, Net Debt/Asset (%) -5.79, Maturity 0.56
  - Firms Exposed to Business Risk: Gross Debt/Asset (%) 43.37, Net Debt/Asset (%) 23.75, Maturity 0.75
- Baseline OLS regression results (Table 2):
  - Post coefficient on Leverage: -0.053 ∗∗∗ (standard error (0.005))
  - Post coefficient on Long-term Leverage: -0.026 ∗∗∗ (standard error (0.003))
  - Post coefficient on Debt Maturity: 0.015 ∗∗∗ (standard error (0.004))
  - Interpretation: baseline specification suggests leverage ratio decreases by 5.3 percent (pre-shock unconditional mean was 19.56 percent); long-term leverage declines by 2.6 percent; average maturity of outstanding debt increases conditional on controls.
- Difference-in-differences estimates (Table 4):
  - Post × Financial Risk: coefficients around -0.039 ∗∗∗ to -0.024 ∗∗∗ depending on specification — combined with Post effect implies total reduction for firms exposed to financial risk of approximately 8.5 percent.
  - Post × Business Risk: coefficients small and generally not statistically different from zero.

### Robustness checks
- Alternate leverage measure (Gross Debt/Total Assets) yields post-COVID decline of -0.030 (standard error (0.004)); R-squared = 0.052; N = 42822.
- Adding log stock price: post = -0.032 (∗∗∗) (0.006); N = 38475.
- Adding cash ratio: post = -0.023 (∗∗∗) (0.006); N = 35436.
- Restricting pre-COVID window to 2019:Q1-2020:Q1: post = -0.046 (∗∗∗) (0.007); N = 23868.
- Placebo tests assigning Post = 1 to pre-COVID windows yield small and insignificant estimates for leverage and long-term leverage.

### Data and sample characteristics
- Sample period: 2018:Q1 through 2020:Q4; firm-quarter observations from U.S. Compustat Quarterly Database.
- After filters and winsorization: approximately 40,000 firm-quarter observations and approximately 3,100 unique firms.
- Exclusions: utilities and financial services; firms with negative asset value; firms with no debt in any firm-quarter during the sample period.
- Firm size statistics:
  - Median total assets: USD 1.3 billion.
  - Mean total assets: USD 9.48 billion.
  - 75th percentile total assets: USD 5.53 billion.
- Profitability proxy: median EBITDA margin is 15.46 percent.

### Structural model of optimal capital structure (estimation and calibrated parameters)
- Framework: Leland and Toft (1996) model with endogenous default; asset value A_t follows geometric Brownian motion.
- Calibrated parameters (Table 7, Panel A):
  - Risk-free Rate (%) — Pre-COVID: 2.5; Post-COVID: 1.3
  - Tax Rate (%) — Pre-COVID: 30; Post-COVID: 30
  - Payout Rate (%) — Pre-COVID: 0.5; Post-COVID: 0.5
  - Bankruptcy Cost — Pre-COVID: 0.05; Post-COVID: 0.05
- Estimated asset volatility σ_a (Table 7, Panel B mean (Std. Dev)):
  - σ_a, full sample — Pre-COVID: 0.31 (2.39); Post-COVID: 0.35 (2.31)
  - σ_a, firms exposed to FR — Pre-COVID: 0.42 (3.06); Post-COVID: 0.52 (9.1)
  - σ_a, firms exposed to BR — Pre-COVID: 0.32 (1.3); Post-COVID: 0.29 (0.67)
- Maturity estimates (Table 7):
  - Maturity, full sample — Pre-COVID: 7.68 (2.88); Post-COVID: 7.9 (2.5)
  - Maturity, firms exposed to FR — Pre-COVID: 5.66 (2.9); Post-COVID: 6.0 (2.55)
  - Maturity, firms exposed to BR — Pre-COVID: 7.92 (2.7); Post-COVID: 7.76 (2.6)

### Model-implied optimal leverage and over-leveraging (Table 8)
- Optimal leverage (%) — Pre-Covid / Post-Covid / Over-Leveraged:
  - Full Sample: 31.50 / 18.73 / No
  - Firms Exposed to Financial Risk: 17.60 / 8.30 / No
  - Firms Exposed to Business Risk: 31.10 / 19.16 / Yes
- Interpretation:
  - Model predicts decline in value-maximizing leverage of approximately 13 percent (from 31.50 to 18.73 percent), driven primarily by reductions in expected growth rate of cash flows and a spike in asset return volatility.
  - Many firms exposed to business risk are classified as over-leveraged (actual leverage exceeds model-implied optimal leverage).

### Structural default risk estimation and Distance-to-Default (DTD)
- Method: Merton (1974) contingent-claims framework with iterative algorithm (Vassalou and Xing (2004)) to infer asset value and asset volatility; default barrier = Short Term Debt + 0.5 * Long Term Debt − Cash and Cash Equivalents.
- Convergence criterion: |σ_A^n − σ_A^(n−1)| < ε with ε < 0.001.
- Selected DTD estimates (Table 9, Panel A and B exact entries):
  - Panel A: All Firms
    - Pre-COVID: p(50) = 9.76; p(25) = 1.76
    - Post-COVID: p(50) = 10.51; p(25) = 1.40
    - Δ(Post − Pre): 0.75; -0.36
  - Panel B: Large Firms (average total assets > USD 1 billion)
    - Pre-COVID: All Firms p(50) = 10.09; p(25) = 1.90
    - Post-COVID: All Firms p(50) = 10.77; p(25) = 1.52
    - Δ(Post − Pre): 0.68; -0.38
  - Large Firms with BR: Pre-COVID p(50) = 10.2; p(25) = 2.37; Post-COVID p(50) = 9.14; p(25) = 1.22; Δ = -1.06; -1.15
- Cross-sectional/tail dynamics:
  - Median firm DTD rises by about 10 percent from 9.76.
  - Firms in the 25th percentile decline from 1.76 to 1.40 (deterioration by around 20.5 percent).
  - Largest deterioration concentrated among firms exposed to business risk and among the largest firms.

### Stress tests and simulated shocks (Section 6.3 and Table 10)
- Two counterfactual stress tests applied in addition to COVID effects:
  - Shock 1: reduce estimated drift rate μ by 20 percent (simulated 20 percent decline in expected growth rate).
  - Shock 2: raise estimated volatility σ by 20 percent (simulated 20 percent rise in cash flow return volatility).
- Selected results (Table 10 exact entries):
  - Shock 1 — Panel A2 Large Firms:
    - Firms with BR: Pre-COVID p(50) = 8.57, p(25) = 2.01; Post-COVID p(50) = 7.45, p(25) = 0.86; Δ = -1.12, -1.15
    - Interpretation: 25th-percentile large firms with business risk decline to p(25) = 0.86 — asset value less than one standard deviation away from the default barrier.
  - Shock 2 — Panel B2 Large Firms:
    - Firms with BR: Pre-COVID p(50) = 8.47, p(25) = 1.72; Post-COVID p(50) = 7.58, p(25) = 0.89; Δ = -0.89, -0.83
    - Interpretation: similar amplification of vulnerability from higher volatility; bottom quartile of large business-risk firms approach insolvency thresholds under stress.
- Aggregate simulated patterns: bottom quartile of firms (sorted on DTD) — especially large firms exposed to social distancing — are particularly vulnerable to additional adverse shocks.

### Policy-relevant implications and synthesis
- COVID-19 shock lowered model-implied optimal leverage substantially via worsened expected growth and higher asset volatility.
- De-leveraging observed empirically is concentrated among firms with high rollover/financial risk; firms most affected by social distancing generally maintained or increased gross debt and shortened maturity, becoming over-leveraged relative to new optimal levels.
- Over-leveraging and concentrated deterioration in credit risk among large firms and the bottom DTD quartile have implications for financial stability and systemic risk buildup.
- Stress tests indicate that a 20 percent decline in expected growth or a 20 percent rise in volatility would place many bottom-quartile large business-risk firms within one standard deviation of the default barrier.

*Source: Excerpt from wpiea2021265-print-pdf (IMF working paper content provided).*

### 5.3  percentage points  from  the  pre-shock

### 5.3  percentage points  from  the  pre-shock

### Key empirical findings
- Firm leverage (Net Debt/Asset) declined by 5.3 percentage points due to Covid-related lockdowns.
- Pre-crisis mean leverage in the sample is 19.6 percent.
- Mean Gross Debt/Asset in the pre-COVID sample is 33.43 percent.
- Debt maturity increased moderately following the onset of COVID-19.
- De-leveraging and rise in maturity are stronger among firms exposed to high financial (rollover) risk.
- Firms most vulnerable to social distancing did not reduce leverage relative to less-vulnerable firms, although they shortened their debt maturity structure.
- Median leverage ratio rose from 10 percent in early 2007 to 21 percent in 2019:Q4, followed by a decline beginning 2020:Q1.
- U.S. GDP dropped by 9.5 percent quarter-over-quarter (noted in context of the shock).

### Heterogeneity and risk channels
- Financial risk (rollover risk) measured via the current portion of long-term debt: firms with higher refinancing needs reduced leverage by approximately 4 percentage points more than less-exposed firms.
- Business risk (exposure to social distancing) measured by growth in sales in 2020:Q2 (bottom quartile classified as exposed): these firms did not exhibit additional de-leveraging, and some became over-leveraged relative to model-implied optimal leverage.
- The de-leveraging effect is robust to alternate measures of leverage (Gross Debt/Asset) and liquidity variations.

### Structural model and optimal leverage
- A standard structural model of optimal capital structure (Leland and Toft (1996) framework) is estimated separately for pre-COVID and post-COVID samples.
- Model-implied optimal leverage decreased by 13 percent, driven primarily by:
  - significant reductions in expected growth rate of cash flows, and
  - a spike in asset return volatility.
- Firms that did not de-lever (notably those exposed to business risk) became over-leveraged when actual leverage exceeded model-implied optimal leverage.

### Default risk, distance-to-default (DTD), and stress tests
- Structural estimates of distance-to-default reveal the sharpest deterioration in credit risk for firms that became over-leveraged (those exposed to high business risk).
- Deterioration in credit risk is concentrated among firms in the bottom quartile (sorted on DTD) and among the largest firms.
- Among large firms (average size at least USD 1 billion) exposed to business risk:
  - 25th percentile value of DTD is 1.22, the lowest in examined subsamples.
  - This subset experienced a 50 percent deterioration in distress risk.
- Stress test scenarios (applied to all firms):
  - (i) a 20 percent decline in expected growth rate of cash flows, and
  - (ii) a 20 percent rise in cash flow return volatility.
- Under the stress scenarios, among companies exposed to business risk the 25th percentile DTD is 0.86 — i.e., the asset value of the bottom quartile of these firms is less than one standard deviation away from reaching insolvency.
- Additional stress tests predict value of these firms will be less than one standard deviation away from default if cash flows decline by 20 percent.

### Data and sample characteristics
- Sample period begins 2018:Q1 and uses firm-quarter level data from the U.S. Compustat Quarterly Database through 2020:Q4.
- After filters and winsorization, the sample contains approximately 40,000 firm-quarter observations and approximately 3,100 unique firms.
- Firms excluded: utilities and financial services; firms with negative asset value; firms with no debt in any firm-quarter during the sample period.
- Firm size statistics:
  - Median total assets: USD 1.3 billion.
  - Mean total assets: USD 9.48 billion.
  - 75th percentile total assets: USD 5.53 billion.
- Profitability proxy (EBITDA scaled by operating revenue): median EBITDA margin is 15.46 percent.

### Methodology summary
- Empirical approach: difference-in-differences regressions controlling for firm size, asset tangibility, growth prospects, profitability, and time-invariant firm fixed effects.
- Identification of financial risk: exogenous variation in firms’ need to rollover the current portion of long-term debt (following Almeida et al. (2009)).
- Identification of business risk: unanticipated drop in sales in 2020:Q2; robustness checks use Return on Assets.
- Structural estimation:
  - Optimal leverage estimated by minimizing squared deviation between predicted equity volatility and actual volatility in the Leland and Toft (1996) model.
  - Default probabilities estimated using a canonical Merton (1974) model with an iterative algorithm (Vassalou and Xing (2004)) to infer unobservable asset drift and volatility.

### Implications highlighted in the text
- The COVID-19 shock reduced firms’ optimal leverage through worsened expected growth and higher asset volatility.
- Firms that did not adjust leverage in line with the new optimal levels — notably those exposed to business risk — became over-leveraged and experienced larger increases in default risk.
- Over-leveraging and concentrated deterioration in credit risk among large firms and the bottom DTD quartile have implications for financial stability and systemic risk buildup.

*Source: Excerpt from wpiea2021265-print-pdf (IMF working paper content provided).*

### 2.2  Empirical Fact: Rising Trend in Leverage post-GFC

### 2.2  Empirical Fact: Rising Trend in Leverage post-GFC

### Rising leverage trend (Figure 1)
- Median Net Debt/Asset rose from 10 percent in 2007:Q1 to 22 percent in 2020:Q1.
- Mean leverage exhibits a similar upward trend, marginally less pronounced, with cyclical behavior since the GFC.
- Both mean and median dropped sharply beginning 2020:Q1, co-inciding with the onset of the pandemic; the drop in mean leverage is perhaps more pronounced than previous cycles of de-leveraging.
- Leverage ratio is defined as Net Debt/Asset. Data are winsorized at the 1 and 99 percent level and aggregated across firms by each year-quarter.

### Empirical strategy (Section 2.3)
- Baseline specification (observations at firm-quarter level):
  - Dependent variables: (i) Net Debt/Asset, (ii) Net Long-term Debt/Asset, (iii) Long-term Debt/Total Debt (to proxy maturity).
  - Controls (lagged by one quarter): firm size proxied by log(Assets), Tangibility, Market/Book value of Equity, Profitability proxied by EBIT/Asset.
  - Fixed effects: firm 훼i, year 훿y, sector-time 휆jt.
  - Variable of interest: Post t = 1 for 2020:Q2-2020:Q4. A negative coefficient on Post t implies a decline in leverage following the recessionary shock.
- Defining risk exposures:
  - Business risk (social distancing): firms in the lowest quartile of sales growth in 2020Q2 relative to 2019Q2 classified as exposed to “business risk”.
  - Robustness: alternative business-risk classification uses change in ROA (EBITDA/Total Assets); firms with change in ROA in the lowest quartile in 2020Q2 relative to 2019Q2 are classified as exposed to high Business Risk.
  - Financial risk (rollover/refinancing): firms sorted by ratio of long-term debt due in one year over total debt; firms in the top quartile are classified as exposed to financial risk.
    - The cutoff at the 75th percentile is 20.1 percent meaning companies are exposed to financial risk if their share of the current portion of long-term debt in total debt is 20.1 percent or higher.
- Difference-in-differences specification:
  - Yit = β1 Post t + β2 Post t × Exposed i + γ′ X i,t−1 + α i + δ y + λ jt + ε it.
  - Coefficient of interest: β2, the DiD estimate of causal impact of exposure on capital structure post-COVID.

### Benchmark results (Section 3.1, Table 1 and Table 2)
- Sample means (Table 1):
  - Panel A: Pre-COVID (2018:Q1-2020:Q1)
    - Full Sample: Gross Debt/Asset (%) 33.43, Net Debt/Asset (%) 19.56, Maturity 0.74
    - Firms Exposed to Financial Risk: Gross Debt/Asset (%) 29.28, Net Debt/Asset (%) 2.92, Maturity 0.52
    - Firms Exposed to Business Risk: Gross Debt/Asset (%) 38.65, Net Debt/Asset (%) 23.32, Maturity 0.77
  - Panel B: Post-COVID Onset (2020:Q2-2020:Q4)
    - Full Sample: Gross Debt/Asset (%) 33.91, Net Debt/Asset (%) 16.99, Maturity 0.77
    - Firms Exposed to Financial Risk: Gross Debt/Asset (%) 26.11, Net Debt/Asset (%) -5.79, Maturity 0.56
    - Firms Exposed to Business Risk: Gross Debt/Asset (%) 43.37, Net Debt/Asset (%) 23.75, Maturity 0.75
  - Panel C: Share of Assets by Risk-type (% of total assets)
    - Firms exposed to Financial Risk: 6.09
    - Firms exposed to Business Risk: 10.48
- Key observations:
  - Gross Debt/Asset does not reveal significant de-leveraging overall; Net Debt/Asset reveals significant reduction in leverage post-COVID in firms exposed to financial risk.
  - Firms exposed to financial risk hold large amounts of cash, explaining divergence between gross and net debt measures.
  - Firms exposed to business risk compose 10.5 percent of total assets in U.S. publicly listed firms (Panel C).
- Baseline OLS regression results (Table 2, Columns (1)-(3)):
  - Post coefficient on Leverage: -0.053 ∗∗∗ (standard error (0.005))
  - Post coefficient on Long-term Leverage: -0.026 ∗∗∗ (standard error (0.003))
  - Post coefficient on Debt Maturity: 0.015 ∗∗∗ (standard error (0.004))
  - R2: 0.039, 0.039, 0.018 respectively.
  - N: 385204, 165039, 257
  - Interpretation: baseline specification suggests leverage ratio decreases by 5.3 percent (pre-shock unconditional mean was 19.56 percent); long-term leverage declines by 2.6 percent; average maturity of outstanding debt increases conditional on controls.
- Comparison with prior recessions:
  - Reduction of 1.5 percent in leverage following the global financial crisis.
  - Insignificant change in leverage following the dotcom bubble.

### Placebo and robustness (Table 3)
- Placebo test: drop post-COVID sample, randomly assign Post = 1 to pre-COVID windows (2019:Q3-2020:Q1 and 2019:Q4-2020:Q1).
- Results: small and insignificant estimates for leverage and long-term leverage in placebo tests; no decline in leverage detected in pre-COVID randomly assigned samples.
- Note: a significant coefficient on maturity in one placebo iteration suggests some pre-COVID rising trend in maturity may exist.

### Risk-exposure heterogeneity (Table 4 and Table 5)
- Difference-in-differences on leverage (Table 4):
  - Post: -0.055 ∗∗∗ (0.006) in Column (1); similar negative Post coefficients across columns.
  - Post × Business Risk: 0.010 (0.010) / 0.012 (0.010) / 0.010 (0.006) / 0.010 (0.006) — generally small and not statistically different from zero.
  - Post × Financial Risk: -0.039 ∗∗∗ (0.010) / -0.037 ∗∗∗ (0.010) / -0.025 ∗∗∗ (0.007) / -0.024 ∗∗∗ (0.007) — economically and statistically significant negative effects for firms exposed to financial risk.
  - R2 ranges around 0.037–0.042. N varies by specification (e.g., 36268, 36538, 34838, 39379, 39486, 37773).
  - Interpretation: firms exposed to financial/rollover risk reduced leverage substantially. Combined with Post effect, total reduction in leverage for firms exposed to financial risk is approximately 8.5 percent (sum of the relevant coefficients reported).
  - Firms exposed to business risk did not systematically change debt levels relative to less-exposed firms; raw means suggest no de-leveraging and Gross Debt/Asset may have increased for business-risk firms.
- Debt maturity (Table 5):
  - Post: 0.026 ∗∗∗ (0.004) in Column (1); other columns show 0.002 (0.004) and 0.011 ∗∗ (0.005).
  - Post × Business Risk: -0.038 ∗∗∗ (0.007) and -0.030 ∗∗∗ (0.007) — firms exposed to business risk lowered their maturity structure.
  - Post × Financial Risk: 0.034 ∗∗∗ (0.009) and 0.037 ∗∗∗ (0.009) — firms exposed to financial risk raised their maturity structure.
  - R-squared: 0.020, 0.012, 0.014. N: 37733, 33847, 32480.
  - Interpretation: firms with rollover/financial risk lengthened debt maturity (consistent with Diamond and He (2014)); firms most affected by social distancing shortened maturity, consistent with Leland and Toft (1996) or possible interest-cost minimization.
- Sector patterns:
  - Figure 5 and referenced plots show sectors most affected by social distancing (e.g., Education, Entertainment) lowered their maturity structure between 2019Q2 and 2020Q2.

*Source: wpiea2021265-print-pdf - 2.2  Empirical Fact: Rising Trend in Leverage post-GFC*

### 3.3  Robustness Tests

### 3.3  Robustness Tests

### Alternate Measures of Leverage
- Table 6, column (1): using Gross Debt/Total Assets (gross debt defined as sum of long-term and short-term debt scaled by assets) in the baseline pre-post specification yields a post-COVID decline in gross debt scaled by asset of 0.03 푝표푠푡 (-0.030) with standard error (0.004). R-squared = 0.052. Firm FE = Y, Year FE = Y, Sector x Quarter FE = Y, Controls = Y, Additional Controls = N. N = 42822.
- Interpretation: The impact on leverage is not sensitive to the choice of debt measure; gross-debt measure shows a 3 percentage point decline post-COVID.

### Control Variables
- Table 6, column (2): adding log of quarterly closing stock price yields 푝표푠푡 = -0.032 (∗∗∗) with standard error (0.006). R-squared = 0.053. Firm FE = Y, Year FE = Y, Sector x Quarter FE = Y, Controls = Y, Additional Controls = Y. N = 38475.
- Table 6, column (3): adding cash ratio (defined in Table A1 in the Appendix) yields 푝표푠푡 = -0.023 (∗∗∗) with standard error (0.006). R-squared = 0.124. Firm FE = Y, Year FE = Y, Sector x Quarter FE = Y, Controls = Y, Additional Controls = Y. N = 35436.
- Interpretation: Controlling for valuation effects (log stock price) and corporate liquidity (cash ratio) leaves the qualitative result unchanged; point estimates are nearly identical or marginally smaller but remain statistically significant.

### Sample Time-Frame
- Table 6, column (4): restricting the pre-COVID window to 2019Q1-2020:Q1 yields 푝표푠푡 = -0.046 (∗∗∗) with standard error (0.007). R-squared = 0.037. Firm FE = Y, Year FE = Y, Sector x Quarter FE = Y, Controls = Y, Additional Controls = N. N = 23868.
- Interpretation: Using 2019:Q1-2020:Q1 as the pre-COVID window produces a point estimate quite similar to the benchmark, reducing concern that unobserved longer-run time-series variation drives results.

### Alternate Measure for Business Risk from Social Distancing
- Using Return on Assets as the exposure measure to business risk from social distancing leaves primary results unchanged. (Regressions available upon request.)

### Notes on Table 6 (alternate tests on leverage)
- Standard errors are clustered at the firm level.
- Dependent variable in column (1): gross debt ratio (long-term + short-term debt scaled by assets).
- Column-level sample sizes: N = 42822, 38475, 35436, 23868 respectively.

---

### Model of Optimal Capital Structure (summary of subsequent sections included in the content)
- Model framework: structural model of optimal capital structure following Leland and Toft (1996) with endogenous default; all agents are risk-neutral; asset value A_t follows geometric Brownian motion as specified in equation (3).
- Firm value decomposition: v(A) = A + TB(A) − BC(A) as in equation (4) with tax-deductible coupons at rate τ and bankruptcy cost fraction α of firm value.
- Equity and debt valuations defined by equations (5) and (6); default A_B chosen endogenously to maximize equity value per (7).

### Estimation Strategy (key calibration and estimation choices)
- Calibrated parameters:
  - Risk-free Rate (r): Pre-COVID 2.5 percent, Post-COVID 1.3 percent (20-year Risk-free Rate).
  - Tax Rate (τ): 30 percent.
  - Payout Rate (δ): 0.5 percent.
  - Bankruptcy Cost (α): 0.05 (Goldstein et al. (2001)).
- Asset volatility modeled as σ_a = exp(β_0 + n'β_i X_i) (equation (8)), estimated via maximum likelihood; equity volatility σ_e related to σ_a via transformation l(σ_a).
- Debt maturity parameterized: T = 1 + 9*(Long term Debt / Total Debt) (equation (9)).
- Face value of debt (P) proxied by net book leverage; drift proxied by long-term risk-free rate; payout rate δ = 0.5 percent.
- Estimation performed separately for pre and post-COVID samples for: (i) all firms, (ii) firms exposed to financial risk (FR), (iii) firms exposed to business risk from social distancing (BR).
- Predicted σ_a and maturities reported in Table 7.

### Table 7: Calibrated and Estimated Parameter Values (selected exact entries)
- Panel A: Calibrated Values
  - Risk-free Rate (%) — Pre-COVID: 2.5; Post-COVID: 1.3
  - Tax Rate (%) — Pre-COVID: 30; Post-COVID: 30
  - Payout Rate (%) — Pre-COVID: 0.5; Post-COVID: 0.5
  - Bankruptcy Cost — Pre-COVID: 0.05; Post-COVID: 0.05
- Panel B: Estimated Values (Mean (Std. Dev))
  - σ_a, full sample — Pre-COVID: 0.31 (2.39); Post-COVID: 0.35 (2.31)
  - σ_a, firms exposed to FR — Pre-COVID: 0.42 (3.06); Post-COVID: 0.52 (9.1)
  - σ_a, firms exposed to BR — Pre-COVID: 0.32 (1.3); Post-COVID: 0.29 (0.67)
  - Maturity, full sample — Pre-COVID: 7.68 (2.88); Post-COVID: 7.9 (2.5)
  - Maturity, firms exposed to FR — Pre-COVID: 5.66 (2.9); Post-COVID: 6.0 (2.55)
  - Maturity, firms exposed to BR — Pre-COVID: 7.92 (2.7); Post-COVID: 7.76 (2.6)

### Model-Implied Optimal Leverage (Table 8 exact figures and classification)
- Optimal leverage (%) — Pre-Covid / Post-Covid / Over-Leveraged
  - Full Sample: 31.50 / 18.73 / No
  - Firms Exposed to Financial Risk: 17.60 / 8.30 / No
  - Firms Exposed to Business Risk: 31.10 / 19.16 / Yes
- Interpretation and links to observed leverage:
  - Observed leverage (recall Table 1): pre-COVID 19.56 percent; post-COVID around 16.99 percent for the full sample.
  - Model prediction: decline in value-maximizing leverage of approximately 13 percent (from 31.50 to 18.73 percent) driven by rise in estimated volatility and a sharp decline in expected growth rate (proxied by long-term risk-free rate), with greater impact attributable to decline in expected growth rate.
  - Firms exposed to financial risk: model predicts ~50 percent decline in optimal leverage (driven by large increase in estimated firm risk).
  - Firms exposed to business risk: model documents a large decline in optimal leverage to around 18.73 percent post-COVID; many such firms are classified as over-leveraged (actual leverage exceeds model-implied optimal leverage), implying higher corporate fragility and systemic risk.

### Structural Estimation of Default Probability (methodology and selected results)
- Method: Merton (1974) contingent-claims framework using Black and Scholes (1973) formula linking equity to asset value (equation (10)); iterative algorithm following Vassalou and Xing (2004) and Bharath and Shumway (2008) to infer asset value and asset volatility.
- Default barrier D defined as: Short Term Debt + 0.5 * Long Term Debt − Cash and Cash Equivalents (equation (15)).
- Initial guess for σ_A: σ_A^0 = σ_E * (E / (E + D)); initial drift proxied by 10-year constant treasury rate.
- Convergence criterion: |σ_A^n − σ_A^(n−1)| < ε with ε < 0.001.
- Distance-To-Default (DTD) and default probability formulas:
  - DTD as in equation (17).
  - P_def = N(−DTD) as in equation (18).
- Aggregation: estimates averaged unweighted by year-quarter to produce time-series.

### Structural Model Estimates: Default Risks Concentrated Among Larger Over-Leveraged Firms
- Time-series behavior (Figure 6 summary): default probabilities elevated around the financial crisis, dip around 2009, sharp increases coinciding with Eurozone sovereign debt crisis and the sharpest single-quarter increase in 2020 at COVID onset.
- Table 9: Distance-To-Default Estimates (exact table entries)

  Panel A: All Firms
  - Pre-COVID: p(50) = 9.76; p(25) = 1.76
  - Post-COVID: p(50) = 10.51; p(25) = 1.40
  - Δ(Post − Pre): 0.75; -0.36

  Panel A (Firms with FR)
  - Pre-COVID: p(50) = 7.76; p(25) = 1.43
  - Post-COVID: p(50) = 8.83; p(25) = 1.33
  - Δ(Post − Pre): 1.07; -0.10

  Panel A (Firms with BR)
  - Pre-COVID: p(50) = 9.34; p(25) = 1.88
  - Post-COVID: p(50) = 9.39; p(25) = 1.51
  - Δ(Post − Pre): 0.05; -0.37

  Panel B: Large Firms (average total assets > USD 1 billion)
  - Pre-COVID: All Firms p(50) = 10.09; p(25) = 1.90
  - Post-COVID: All Firms p(50) = 10.77; p(25) = 1.52
  - Δ(Post − Pre): 0.68; -0.38

  Panel B (Large Firms with FR)
  - Pre-COVID: p(50) = 11.97; p(25) = 3.86
  - Post-COVID: p(50) = 13.15; p(25) = 2.81
  - Δ(Post − Pre): 1.18; -1.05

  Panel B (Large Firms with BR)
  - Pre-COVID: p(50) = 10.2; p(25) = 2.37
  - Post-COVID: p(50) = 9.14; p(25) = 1.22
  - Δ(Post − Pre): -1.06; -1.15

- Notes: DTD estimates use assumed maturity = 1 year; default barrier = short-term debt + 0.5*long-term debt − cash and cash equivalents; risk-free rate proxied by 10-year constant maturity treasury rate.

_Italic: Source — wpiea2021265-print-pdf (sections 3.3 through parts of 6 as provided)._

### 9. We rely on the DTD metric widely used in the credit risk literature. Column 1 in Panel

### wpiea2021265-print-pdf - 9. We rely on the DTD metric widely used in the credit risk literature. Column 1 in Panel

### Distance-to-Default (DTD) findings: cross-sectional and tail dynamics
- Median firm DTD rises by about 10 percent from 9.76 (median moves further away from default).
- Firms in the 25th percentile (bottom quartile) experience a decline in DTD from 1.76 to 1.40, a deterioration by around 20.5 percent.
- Firms exposed to business risk show the largest change in Column (6): a decline of 0.37 to 1.51.
- Patterns are amplified for firms with asset value ≥ USD 1 billion (Large firms):
  - Firms in the 25th percentile in the Large sample experienced a decline of 0.38 (as reported).
  - For firms exposed to social distancing (business risk), the 25th percentile DTD is 1.22, the lowest across reported sub-samples.
- Interpretation: stressed firms pre-COVID became more vulnerable to bankruptcy post-COVID; large firms exposed to social distancing became much closer to default.

### Stress tests (Section 6.3): design and simulated shocks
- Two counterfactual stress tests applied "over and above" the COVID recession effects:
  - Shock 1: reduce each company’s estimated drift rate, μ, by 20 percent (simulated 20 percent decline in expected growth rate).
  - Shock 2: raise each company’s estimated volatility, σ, by 20 percent (simulated 20 percent rise in cash flow return volatility).
- Simulations re-estimate DTD pre- and post-COVID, accounting for heterogeneous leverage adjustments across firms.
- Notes on methodology (as reported):
  - Asset volatilities and implied asset values come from structural model estimates.
  - Assumed maturity for estimating asset volatilities is 1 year.
  - Default barrier = short-term debt + half of long-term debt − cash and cash equivalents.
  - Risk-free rate proxied by the 10-year constant maturity treasury rate.
  - Large firms defined as average total asset value greater than USD 1 billion.

### Table 10: Key DTD statistics (stress tests) — selected values preserved exactly
- Shock 1: Growth Slowdown
  - Panel A1: All Firms
    - Pre-COVID p(50) = 7.96, p(25) = 1.44
    - Post-COVID p(50) = 8.76, p(25) = 1.01
    - Δ(Post−Pre) p(50) = 0.80, p(25) = -0.43
    - Firms with FR: Pre-COVID p(50) = 6.95, p(25) = 1.20; Post-COVID p(50) = 7.23, p(25) = 1.01; Δ = 0.28, -0.19
    - Firms with BR: Pre-COVID p(50) = 7.82, p(25) = 1.53; Post-COVID p(50) = 7.95, p(25) = 1.06; Δ = 0.13, -0.47
  - Panel A2: Large Firms
    - Pre-COVID p(50) = 8.39, p(25) = 1.56
    - Post-COVID p(50) = 9.01, p(25) = 1.11
    - Δ(Post−Pre) p(50) = 0.62, p(25) = -0.45
    - Firms with FR: Pre-COVID p(50) = 10.34, p(25) = 3.19; Post-COVID p(50) = 11.13, p(25) = 2.22; Δ = 0.79, -0.97
    - Firms with BR: Pre-COVID p(50) = 8.57, p(25) = 2.01; Post-COVID p(50) = 7.45, p(25) = 0.86; Δ = -1.12, -1.15
    - Interpretation: Panel A2 shows amplified simulated effects for larger firms; 25th-percentile large firms with rollover risk decline by 0.97 to 2.22, and with business risk decline by 1.15 to 0.86 — indicating some relatively large firms are over-leveraged with asset value less than one-standard deviation away from the default barrier.
- Shock 2: Higher Uncertainty (Asset Volatility)
  - Panel B1: All Firms
    - Pre-COVID p(50) = 7.86, p(25) = 1.31
    - Post-COVID p(50) = 8.73, p(25) = 1.03
    - Δ(Post−Pre) p(50) = 0.87, p(25) = -0.28
    - Firms with FR: Pre-COVID p(50) = 6.43, p(25) = 0.97; Post-COVID p(50) = 7.30, p(25) = 0.96; Δ = 0.87, -0.01
    - Firms with BR: Pre-COVID p(50) = 7.74, p(25) = 1.35; Post-COVID p(50) = 7.79, p(25) = 1.17; Δ = 0.05, -0.18
  - Panel B2: Large Firms
    - Pre-COVID p(50) = 8.38, p(25) = 1.41
    - Post-COVID p(50) = 8.96, p(25) = 1.11
    - Δ(Post−Pre) p(50) = 0.58, p(25) = -0.30
    - Firms with FR: Pre-COVID p(50) = 9.94, p(25) = 3.16; Post-COVID p(50) = 10.94, p(25) = 2.28; Δ = 1.00, -0.88
    - Firms with BR: Pre-COVID p(50) = 8.47, p(25) = 1.72; Post-COVID p(50) = 7.58, p(25) = 0.89; Δ = -0.89, -0.83
  - Interpretation: Very similar patterns to growth-shock stress test, reinforcing that a 20 percent spike in cash flow volatility raises vulnerability of over-leveraged firms.

### Aggregate conclusions and policy implications (Section 7 and synthesis)
- Empirical estimates (data through 2020Q4) indicate COVID shock decreased leverage by 5.3 percent relative to the pre-shock mean of 19 percent, while debt maturity increased marginally.
- De-leveraging results are driven mostly by firms exposed to significant rollover risk pre-COVID; firms affected by social distancing generally maintained leverage ratios and shortened debt maturity.
- Structural-model estimation of drift and volatility pre- and post-COVID shows a significant reduction in optimal corporate leverage levels.
- Firms most severely affected by social distancing chose to maintain leverage and thus became over-leveraged; structural credit-risk models confirm these firms experienced the largest deterioration in credit risk.
- Simulation tests show the bottom quartile (sorted on default likelihood) of these firms are particularly vulnerable to future downturns.
- Distress risk is larger for the largest firms affected by social distancing; given large firms’ potential to generate systemic risk, these results have serious implications for financial stability and systemic risk.

*Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021265-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021265-print-pdf.pdf_