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### 1. Introduction — Overview and motivation
- Financial crises often follow periods of asset overvaluation; 2020 equity markets fell sharply at the start of the Covid-19 pandemic then recovered to historical record high levels by end-2020 despite weak macro fundamentals and high vaccine uncertainty.
- High equity valuations do not necessarily imply low firm probability of default (PD); PD depends on market value, macro-financial factors, leverage, profitability, and liquidity.
- Paper proposes a Distance-to-Default to PD (DD-PD) mapping as a simpler empirical alternative to full multi-period, multi-variable structural or forward-intensity scenario models.
- Mapping components:
  - Black-Scholes-Merton (BSM) Distance-to-Default (DD).
  - Forward-intensity PDs from Credit Research Initiative (CRI).
  - Quantile regressions (QR; Koenker and Bassett, 1978) to capture conditional PD distribution and interpret QR lines as risk regimes.

### 2. Anecdotal evidence on equity prices and PDs
- Key macro and market observations:
  - IMF January 2021 projections: world real GDP growth fell to 2.8 percent in 2020 down from 3.5 percent in the previous year.
    - Advanced economies experienced a 5 percent decline.
    - Emerging markets and developing economies declined by 2.4 percent.
    - India recorded a contraction of 8 percent and Latin America a contraction of 7.4 percent.
  - Low interest rate policies since the global financial crisis have been a major driver of equity prices; U.S. episodes with strong stock performance coincide with rates below 1 percent (January 2009–December 2015 and the year 2020).
  - Mean 1-year PD for publicly listed U.S. firms trended upwards since 2014 despite rising market valuations.

### 3. The DD-PD mapping — Building blocks and estimation choices
- Required choices before estimation:
  - Equity-based risk measure: BSM Distance-to-Default (DD).
  - PD database: Credit Research Initiative (CRI), National University of Singapore.
  - Estimation method: Quantile regression (QR).
- Rationale:
  - Collect BSM-DD and corresponding CRI-PDs for all firm observations and use QR to build DD-PD mapping.
  - QR allows associating different QR lines with different risk regimes (high quantiles = high risk regimes).

### 3.1 BSM Distance-to-Default (DD) — Concept and calibration
- Option-pricing representation (as in source):
  - V = A Φ(d1) − D exp(−rT) Φ(d2)
  - d1 = log (A/D) + ( r + 1/2 σ^2 ) T / (σ √T)
  - d2 = d1 − σ √T
  - Variables: A = asset value, D = default barrier, r = risk-free rate, T = time horizon, σ = asset volatility.
- Default barrier D typically set to book value of short-term debt plus half book value of long-term debt.
- A and σ are unobservable; estimated by reverse-engineering V series using iterative Vassalou and Xing (2003) method (algorithm summarized in text).
- DD computation after convergence:
  - DD = log(A/D) + (r − 1/2 σ^2) T / (σ √T).
- Interpretation:
  - If asset value follows geometric Brownian motion with drift = r and volatility = σ, probability asset value < D by time T is Φ(−DD).
  - DD used to rank firms by riskiness; empirical mappings from DD to PDs are applied.

### 3.2 CRI forward intensity-based PDs (CRI-PDs) — Data and model
- CRI database coverage and model:
  - 73,000 publicly listed firms in 133 countries; includes delisted and live firms; dataset goes back to 1990; updated daily for over 36,000 firms.
  - PDs computed using forward-intensity model of Duan, Sun, and Wang (2012).
  - Model uses two independent doubly stochastic Poisson processes (default delisting and other delisting).
  - CRI-PD incorporates sixteen (16) input covariates including macro-financial and firm-specific factors.
- Selected covariates (high-level):
  - Macro-financial: trailing 1-year stock index return; 3-month government bill yield; economy-wide CRI-DD for financial and non-financial firms.
  - Firm-specific: CRI-Distance-to-Default (level and trend); cash-to-total assets (level and trend); current assets to current liabilities (level and trend); net income to total assets (level and trend); relative size (level and trend); relative market-to-book ratio; idiosyncratic volatility.
- Corporate events database:
  - More than 6,000 default events and 43,000 other exit events since 1990 compiled from multiple sources.
  - Default event classes include bankruptcy, missed/delayed payments, debt restructuring/distressed exchanges.
- Calibration groups:
  - Economies grouped into four: North America; Asia-Pacific Developed; Europe; Emerging markets. China and India calibrated separately.
- Prediction performance (CRI model: AR and AUROC by region and horizon) — exact values:
  - China: AR 1 month 0.698, 1 year 0.669, 2 year 0.647, 5 year 0.588; AUROC 1 month 0.849, 1 year 0.836, 2 year 0.826, 5 year 0.801
  - India: AR 1 month 0.730, 1 year 0.694, 2 year 0.670, 5 year 0.592; AUROC 1 month 0.865, 1 year 0.848, 2 year 0.838, 5 year 0.802
  - Developed Asia-Pacific: AR 1 month 0.864, 1 year 0.759, 2 year 0.676, 5 year 0.532; AUROC 1 month 0.932, 1 year 0.879, 2 year 0.839, 5 year 0.768
  - Emerging markets: AR 1 month 0.802, 1 year 0.752, 2 year 0.690, 5 year 0.541; AUROC 1 month 0.901, 1 year 0.876, 2 year 0.846, 5 year 0.774
  - Europe: AR 1 month 0.870, 1 year 0.753, 2 year 0.644, 5 year 0.485; AUROC 1 month 0.935, 1 year 0.877, 2 year 0.823, 5 year 0.745
  - North America: AR 1 month 0.943, 1 year 0.855, 2 year 0.759, 5 year 0.595; AUROC 1 month 0.971, 1 year 0.928, 2 year 0.881, 5 year 0.801
- Observation: Accuracy is better for advanced economies and for shorter forecasting horizons.

### 3.3 Quantile regression (QR) for DD-PD mapping and risk regimes
- Motivation:
  - For a fixed DD, PD can vary widely due to factors not captured directly by DD (profitability, liquidity, cyclical stage).
- QR approach:
  - Estimated quantiles τ = (0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95, 0.99).
  - General linear QR specification:
    - logPD_i = α_0(τ) + α_1(τ) DD_i + X...T β(τ) + ε_i, τ ∈ T, i = 1,...,N.
  - X... denotes cyclical indicators (economy- or sector-specific).
- Baseline risk regime selection:
  - τ_* ≡ arg min_τ { Σ_{i=1}^N (QR_τ(DD_i, X...) − logPD_i)^2 }.
  - Baseline regime can be estimated at firm, portfolio, sector, group of sectors, or economy-wide level.

### 3.4 Scenario analysis framework using DD-PD mapping
- Scenario types:
  - Shocks in the baseline regime (firm remains in current risk regime).
  - Shocks in the stress regime (firm migrates to higher risk regime).
  - Shocks in arbitrary risk regimes (migration to any specified quantile/regime).
- Common scenario inputs:
  - Values of four primary DD parameters: market value of firm, asset volatility, risk-free rate, default barrier.
  - Values of covariates X... used in QR estimation.
  - Choice of risk regime and firm risk-regime migration.
- Algorithms summarized:
  - Algorithm 2 (Baseline risk regime): compute post-shock DD_s, compute fitted QR pre/post-shock, scale observed logPD by ratio of fitted QR predictions.
  - Algorithm 3 (Stress analysis): same steps but use high-risk quantile (e.g., 95th) for post-shock fitted PD.
  - Algorithm 4 (Arbitrary regime): use chosen target quantile (e.g., 25th) for post-shock fitted PD.
- Practical note: Scenario values for DD parameters and covariates can come from empirical models or expert judgment.

### 4–5. Scenario analysis applied to U.S. corporate sector — design and data
- Scenarios considered (exact wording from source):
  - Scenario 1: a broad market correction of and an increase in asset volatility by 10 percent,
  - Scenario 2: a broad market correction of and an increase in asset volatility by 20 percent, and
  - Scenario 3: a broad market correction of and an increase in asset volatility by 30 percent.
  - Stress analysis: stress regime set equal to the 95th quantile regression with current date set to end-December 2020.
  - Note: the price correction in Scenario 3 is similar to the stock market correction observed after the bursting of the 2000 dotcom bubble in the United States.
- Data and estimation specifics:
  - Sample: Over 4600 publicly listed firms (including dead and currently listed firms), sample period August 1985–December 2020.
  - Monthly market value, short-term and long-term liabilities from Thomson Reuters.
  - 1-year PD time series from CRI.
  - 1-year risk free rate = yield of the 1-year U.S. Treasury bill.
  - Asset value and volatility estimated using past 60-month observations (minimum 12 monthly data points allowed).
  - Raw dataset ≈ 9 million observations; processed usable observations ≈ 700,000 observations.
  - Quantile regressions estimated individually for 65 industrial subsectors, aggregated into nine industrial sectors.
  - Cyclical covariates used (all seasonally adjusted): Conference Board Leading Economic Index; Capacity Utilization Index; Consumer Confidence Index; ISM Non-Manufacturers’ Survey Index; ISM Purchasing Managers’ Index; unemployment rate.
  - Quantiles estimated: 5, 10, 25, 50, 75, 90, 95, and 99.

### 5. Results — Asset volatility, leverage, and baseline risk regimes (end-December 2020)
- Asset volatility and leverage:
  - Medium-term volatilities spiked after 1997 and 1998, then trended downwards; spike in PD during 2008 driven mainly by stock market correction rather than increased asset volatility.
  - Quantitative easing pressured medium-term volatility downwards until 2013; since then asset volatility rising in line with 1-year PD.
  - Most volatile sectors at end-December 2020: communications, consumer non-cyclical, energy, and technology.
  - Leverage measured as default barrier (short-term debt + one half long-term debt) divided by asset value.
  - Non-cyclical and technology sectors: high volatility partly offset by lower leverage ratios.
- Baseline risk regimes (selected industry-group quantiles; exact values):
  - Diversified Financial Services: 25
  - Investment Companies: 25
  - Real Estate: 25
  - Savings & Loans: 25
  - Auto Manufacturers: 50
  - Home Builders: 50
  - Textiles: 50
  - Agriculture: 50
  - Household Products/Wares: 50
  - Energy-Alternate Sources: 50
  - Banks: 50
  - Insurance: 50
  - REITS: 50
  - Environmental Control: 50
  - Semiconductors: 50
  - Software: 50
  - Chemicals: 75
  - Forest Products & Paper: 75
  - Iron/Steel: 75
  - Internet: 75
  - Telecommunications: 75
  - Distribution/Wholesale: 75
  - Home Furnishings: 75
  - Housewares: 75
  - Building Materials: 75
  - Electrical Components & Equipment: 75
  - Electronics: 75
  - Machinery-Construction & Mining: 75
  - Metal Fabricate/Hardware: 75
  - Packaging & Containers: 75
  - Transportation: 75
  - Trucking & Leasing: 75
  - Computers: 75
  - Media: 95
  - Retail: 95
  - Toys/Games/Hobbies: 95
  - Healthcare-Services: 95
  - Pharmaceuticals: 95
  - Oil & Gas Services: 95
  - Pipelines: 95
  - Office/Business Equipment: 95
  - Advertising: 99
  - Apparel: 99
  - Office Furnishings: 99
  - Beverages: 99
  - Coal: 99
  - Airlines: 90
  - Auto Parts & Equipment: 90
  - Entertainment: 90
  - Leisure Time: 90
  - Lodging: 90
  - Biotechnology: 90
  - Commercial Services: 90
  - Mining: 90
  - Aerospace/Defense: 90
  - Engineering & Construction: 90
  - Hand/Machine Tools: 90
  - Machinery-Diversified: 90
  - Miscellaneous Manufacturing: 90
  - Water: 90
- Interpretation:
  - Construction, real estate, and financial services groups generally in normal and low risk regimes (quantiles ≤ 50).
  - Many industry subsectors are in high risk regimes; apparel, office equipment, and energy (except green energy companies) are highlighted as particularly high risk.

### 6. Results — Probabilities of Default (PDs) by sector (end-December 2020 and scenarios)
- Liability-weighted PDs (in bps) and scenario outcomes (exact values excerpted):
  - Basic materials: Dec. 2020 = 22; Scenario 1 = 30; Scenario 2 = 48; Scenario 3 = 82; Stress Scenario 1 = 117; Stress Scenario 2 = 161
  - Communications: Dec. 2020 = 27; Scenario 1 = 34; Scenario 2 = 45; Scenario 3 = 59; Stress Scenario 1 = 54; Stress Scenario 2 = 71; Stress Scenario 3 = 90
  - Consumer, cyclical: Dec. 2020 = 63; Scenario 1 = 77; Scenario 2 = 98; Scenario 3 = 124; Stress Scenario 1 = 137; Stress Scenario 2 = 170; Stress Scenario 3 = 206
  - Consumer, non-cyclical: Dec. 2020 = 29; Scenario 1 = 37; Scenario 2 = 51; Scenario 3 = 71; Stress Scenario 1 = 58; Stress Scenario 2 = 93; Stress Scenario 3 = 148
  - Energy: Dec. 2020 = 152; Scenario 1 = 181; Scenario 2 = 226; Scenario 3 = 279; Stress Scenario 1 = 120; Stress Scenario 2 = 151; Stress Scenario 3 = 187
  - Financial: Dec. 2020 = 23; Scenario 1 = 31; Scenario 2 = 42; Scenario 3 = 58; Stress Scenario 1 = 17; Stress Scenario 2 = 82; Stress Scenario 3 = 292
  - Industrial: Dec. 2020 = 25; Scenario 1 = 36; Scenario 2 = 59; Scenario 3 = 93; Stress Scenario 1 = 75; Stress Scenario 2 = 116; Stress Scenario 3 = 168
  - Technology: Dec. 2020 = 17; Scenario 1 = 23; Scenario 2 = 36; Scenario 3 = 55; Stress Scenario 1 = 89; Stress Scenario 2 = 125; Stress Scenario 3 = 169
  - Utilities: Dec. 2020 = 23; Scenario 1 = 36; Scenario 2 = 64; Scenario 3 = 105; Stress Scenario 1 = 112; Stress Scenario 2 = 172; Stress Scenario 3 = 247
- Key observations:
  - In baseline regimes, implied rating downgrades of one to two notches; in stress regimes, up to five notches.
  - Implied rating bands can be wide and mask substantial changes in creditworthiness.
  - Sectoral PD distributions generally shift right and develop fatter right tails under shocks; exception noted for energy under Scenario 1 due to risk-regime migration mechanics.
  - Technology and financial sectors are particularly sensitive to shocks despite technology’s low leverage and financials’ low asset volatility.
  - Cross-sector comparisons limited by differing DD-PD mappings across sectors and subsectors.

### 7. Results — Economic capital impacts for corporate loan portfolios
- Baseline median 1-year PD at end-December 2020 = 10 bps (equivalent to S&P BBB-rating).
  - Global historical 1-year transition rate associated with this rating = 0.16 percent.
- Policy-relevant observation: main systemic concern is the need for lenders to increase provisions and economic capital buffers rather than defaults per se.
- Capital formula used (Basel II ASRF IRB function without maturity adjustment):
  - K = LGD × Φ( √(1−R) × Φ−1(PD) + √(R/(1−R)) × Φ−1(0.999) ) − PD × LGD
  - Parameters: LGD = 40 percent; R = 30 percent.
- Median economic capital requirements reported as multiples of December 2020 values (exact table excerpts):
  - Basic materials: Baseline Scenario 1 = 1.21; Scenario 2 = 1.56; Scenario 3 = 1.88; Stress Scenario 1 = 1.84; Stress Scenario 2 = 2.13; Stress Scenario 3 = 2.32
  - Communications: Baseline 1.17, Scenario 2 1.45, Scenario 3 1.70; Stress 1.68, 1.90, 2.03
  - Consumer, cyclical: Baseline 1.14, 1.35, 1.52; Stress 1.23, 1.42, 1.56
  - Consumer, non-cyclical: Baseline 1.15, 1.41, 1.64; Stress 1.37, 1.59, 1.78
  - Energy: Baseline 1.05, 1.13, 1.19; Stress 1.01, 1.02, 1.06
  - Financial: Baseline 1.29, 1.78, 2.24; Stress 3.17, 3.22, 3.23
  - Industrial: Baseline 1.21, 1.57, 1.92; Stress 1.82, 2.12, 2.31
  - Technology: Baseline 1.28, 1.87, 2.45; Stress 3.23, 3.50, 3.80
  - Utilities: Baseline 1.28, 1.73, 2.14; Stress 2.07, 2.41, 2.65
- Aggregate impacts (exact phrasing and figures):
  - In baseline risk regimes, price shocks of 10 percent, 20 percent, and 30 percent cause median capital requirements to increase on average by 20 percent, 54 percent, and 85 percent respectively.
  - In stress regime, increases are 94 percent, 114 percent, and 130 percent respectively.
  - Energy sector exposures (already in very high risk/stressed regimes) do not require substantial capital increases under these scenarios.

### 8. Conclusions and policy-relevant implications
- The DD-PD mapping links equity-based DD to real-world PDs via quantile regression; each QR line can be associated with a risk regime.
- Uses of the mapping:
  - Identify current risk regime of individual firms, industry groups, and sectors from bottom-up perspective.
  - Perform what-if scenario analysis by simulating shocks to DD determinants under baseline or stressed risk regimes.
  - Applications include managing corporate credit portfolio risk and informing regulatory stress testing.
- Empirical application to U.S. corporate sector:
  - A sustained price correction of 10 percent could cause PDs to rise 30 to 70 percent above end-December 2020 levels; most affected sectors: energy, financial, and technology.
  - Under stress analysis, PDs could be one order of magnitude higher in financial and technology sectors.
- Additional suggested uses (not explored in detail in this unit):
  - Analyze effects of changes in firms’ liability structures on PD and optimal capital structure.
  - Examine impact of monetary policy on firms’ default risk by assessing relationship between equity prices and the risk-free rate and applying the DD-PD mapping.
  - Model potential migration across risk regimes for firms or sectors under secular or cyclical changes in exogenous covariates to guide credit portfolio composition and long-term cost-of-capital projections (e.g., climate change investment scenarios).

*Source: wpiea2021143-print-pdf*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Overview
- Financial crises often follow periods of asset overvaluation, raising concern about a disconnect between asset prices and the economy (Brunnermeier and Oehmke, 2013).
- In 2020, stock markets experienced large price declines at the start of the Covid-19 pandemic but recovered by end-2020 to reach historical record high levels despite dismal economic fundamentals and high uncertainty about a vaccine.
- High equity valuations do not necessarily imply low firm probability of default (PD); PD depends on market value, macro-financial factors, leverage, profitability, and liquidity.
- The paper proposes a Distance-to-Default to PD (DD-PD) mapping as a simpler empirical alternative to full multi-period, multi-variable structural or forward-intensity scenario models for analyzing how market-based measures translate into PDs.

### Motivation and aim
- Estimating multi-period, multi-variable scenarios is complex and resource intensive; small institutions may not have capability to deploy full scenario models.
- The DD-PD mapping:
  - Builds on the Black-Scholes-Merton (BSM) Distance-to-Default (DD) from structural default models (Black and Scholes, 1973; Merton, 1974).
  - Leverages high empirical accuracy of forward-intensity PDs from a production-quality model (Duan, Sun, and Wang, 2012).
  - Uses quantile regressions (QR; Koenker and Bassett, 1978) to capture the distribution of PDs conditional on DD and to interpret QR lines as different risk regimes.
- The mapping facilitates scenario analysis and stress testing without specifying a full suite of macro-financial covariate paths.

### Outline of the paper
- Section 2: Anecdotal evidence on equity prices and PDs in the corporate sector.
- Section 3: Description of three building blocks—structural model (BSM-DD), forward intensity PDs (CRI-PDs), and quantile regression estimation approach—with computational details for latent parameters (asset value and asset volatility).
- Section 4: How to use QR interpretation of DD-PD mapping for scenario analysis, baseline risk regimes, and stress tests.
- Section 5: Application to U.S. corporate sector: evaluates impact of three scenarios on default risk at the sectoral level and potential increases in economic capital faced by lenders.
- Section 6: Conclusions and suggested further uses of the mapping.

---

### 2. Equity Prices and Probability of Default: Anecdotal Evidence

### Key observations
- After large drops in March 2020, stock markets worldwide bounced back strongly; by end-2020 many countries' stock market valuations were at historical record high levels.
- If market valuations alone were used to judge the pandemic’s impact, 2020 would appear to be the shortest recession ever.
- IMF January 2021 projections: world real GDP growth fell to 2.8 percent in 2020 down from 3.5 percent in the previous year.
  - Advanced economies experienced a 5 percent decline.
  - Emerging markets and developing economies declined by 2.4 percent.
  - India recorded a contraction of 8 percent and Latin America a contraction of 7.4 percent.
- Low interest rate policies since the global financial crisis have been a major driver of equity prices; U.S. episodes with strong stock performance coincide with rates below 1 percent (January 2009–December 2015 and the year 2020).
- High equity valuations by end-2020 could have concealed rising corporate sector risk due to increased borrowing and indebtedness enabled by low funding costs; markets responded negatively to limited withdrawal of support measures.
- Mean 1-year PD for publicly listed U.S. firms trended upwards since 2014 despite rising market valuations, indicating higher equity valuations do not imply markets are oblivious to default risk.

---

### 3. The DD-PD Mapping: Building Blocks

### Choices required before estimation
- An appropriate equity-based risk measure.
- A rich PD database.
- An estimation method suitable for constructing the mapping.

### Selected components for the DD-PD mapping
- Equity-based risk measure: Black-Scholes-Merton (BSM) Distance-to-Default (DD).
- PD database: Credit Research Initiative, National University of Singapore (CRI).
- Estimation method: Quantile regression (QR).

### Rationale
- Collect BSM-DD and corresponding PDs for all firm observations and use QR to build the DD-PD mapping.
- QR allows associating different QR lines with different risk regimes (high quantiles = high risk regimes).

---

### 3.1 The Black-Scholes-Merton (BSM) Distance-to-Default (DD)

### Conceptual framework
- Equity viewed as a call option on firm assets; shareholders receive residual value after debtholders.
- Assuming asset value follows geometric Brownian motion, market capitalization V equals the BSM call option price:
  - V = A Φ(d1) − D exp(−rT) Φ(d2)
  - d1 = log (A/D) + ( r + 1/2 σ^2 ) T / (σ √T)
  - d2 = d1 − σ √T
  - Variables: A = asset value, D = default barrier, r = risk-free rate, T = time horizon, σ = asset volatility.

### Practical calibration
- V is observed; r is set equal to a safe asset yield with maturity T (e.g., U.S. Treasury bills or bonds for U.S. firms).
- Default barrier D typically set to book value of short-term debt plus half book value of long-term debt.
- A and σ are unobservable and must be inferred by reverse-engineering the option pricing equation using time series of market capitalizations V0, V1, ..., VN equally spaced by Δt.

### Estimation approach
- Maximum likelihood methods exist (Duan, 1994), but the paper uses the iterative Vassalou and Xing (2003) method for ease of implementation.
- Lando (2004) reports that iterative method results tend to be similar to maximum likelihood.

Algorithm 1: Vassalou-Xing (2003) iterative method
- Data: Market values V0, V1, ..., VN, initial guess σ0, tolerance ε.
- Result: Return asset volatility σ and asset values A0, A1, ..., AN.
- Steps:
  1. Start with σ(n) in iteration n;
  2. While ‖σ(n+1) − σ(n)‖ > ε do:
     - Calculate A0(σ(n)), A1(σ(n)), ..., AN(σ(n)) by reverse-engineering equation (1) with V0, V1, ..., VN and σ(n) as inputs;
     - Estimate σ(n+1) = sqrt( (1 / (N ∆t)) * sum_{i=1}^N ( ln(V_t / V_{t−1}) − (1 / (N ∆t))(ln V_N − ln V_0) )^2 );
  3. End while.

---

*Italicized source attribution line.*

### 4. Upon convergence, setσ=σ

### wpiea2021143-print-pdf - 4. Upon convergence, setσ=σ

### BSM Distance-to-Default (DD) computation
- Once calibrated/estimated variables are available, compute the BSM DD using:
  - DD = log(A/D) + (r − 1/2 σ^2) T / (σ √T).(2)
- Interpretation and use:
  - If asset value follows geometric Brownian motion with drift = risk-free rate, r, and volatility = σ, by time T the probability that asset value is lower than default barrier D is Φ(−DD).
  - DD is primarily used to rank firms by riskiness; empirical mappings from DD to default rates or PDs are then applied.

### CRI forward intensity-based Probabilities of Default (CRI-PDs)
- Data coverage and model:
  - CRI database: 73,000 publicly listed firms in 133 countries; includes delisted and live firms; dataset goes back to 1990; updated daily for over 36,000 firms.
  - PDs computed using forward-intensity model of Duan, Sun, and Wang (2012) (CRI-PD).
  - Model considers two independent doubly stochastic Poisson processes: one for delisting due to default, another for delisting due to other reasons.
  - CRI-PD incorporates sixteen (16) input covariates including macro-financial and firm-specific factors from Thomson Reuters Datastream and Bloomberg Data License Back Office Product.
- CRI PD model covariates (high-level):
  - Macro-financial factors:
    - Stock index return: Trailing 1-year return of the prime stock market
    - Short-term risk-free rate: 3-month government bill yield
    - Economy-wide CRI-DD, financial firms: Median Distance-to-Default of financial/non-financial firms in each economy inclusive of foreign firms listed in the economy’s exchange
    - Economy-wide CRI-DD, non-financial firms
  - Firm-specific attributes:
    - CRI-Distance-to-Default (level and trend): Volatility adjusted leverage based on Merton (1974) with CRI adjustments
    - Cash-to-total assets (level and trend): For financial firms, logarithm of ratio of cash and short-term investments to total assets
    - Current assets to current liabilities (level and trend): For non-financial firms, logarithm of ratio of current assets to current liabilities
    - Net income to total assets (level and trend): Profitability indicator
    - Relative size (level and trend): Logarithm of ratio of firm’s market capitalization to median market capitalization in the economy
    - Relative market-to-book ratio: Measured relative to the economy’s median level
    - Idiosyncratic volatility: 1-year standard deviation of residuals after regressing market capitalization against the economy’s stock return (Shumway, 2001)
- Corporate events database:
  - More than 6,000 default events and 43,000 other exit events since 1990 gathered from Bloomberg, Compustat, CRSP, Moody’s reports, exchange websites, and news.
  - Default event classes include:
    - Bankruptcy filing, receivership, administration, liquidation or other legal impasse to timely settlement of interest/principal
    - Missed or delayed payment of interest/principal excluding grace period delays
    - Debt restructuring/distressed exchange that diminishes financial obligation
- Calibration groups:
  - Economies grouped into four: North America (Canada, United States); Asia-Pacific Developed (Australia, Hong Kong SAR, Japan, Singapore, South Korea, Taiwan, New Zealand); Europe (all European countries); Emerging markets (worldwide). China and India calibrated separately.
- Prediction performance (CRI model: AR and AUROC by region and horizon)
  - Table 2 values exactly as reported:
    - China: AR 1 month 0.698, 1 year 0.669, 2 year 0.647, 5 year 0.588; AUROC 1 month 0.849, 1 year 0.836, 2 year 0.826, 5 year 0.801
    - India: AR 1 month 0.730, 1 year 0.694, 2 year 0.670, 5 year 0.592; AUROC 1 month 0.865, 1 year 0.848, 2 year 0.838, 5 year 0.802
    - Developed Asia-Pacific: AR 1 month 0.864, 1 year 0.759, 2 year 0.676, 5 year 0.532; AUROC 1 month 0.932, 1 year 0.879, 2 year 0.839, 5 year 0.768
    - Emerging markets: AR 1 month 0.802, 1 year 0.752, 2 year 0.690, 5 year 0.541; AUROC 1 month 0.901, 1 year 0.876, 2 year 0.846, 5 year 0.774
    - Europe: AR 1 month 0.870, 1 year 0.753, 2 year 0.644, 5 year 0.485; AUROC 1 month 0.935, 1 year 0.877, 2 year 0.823, 5 year 0.745
    - North America: AR 1 month 0.943, 1 year 0.855, 2 year 0.759, 5 year 0.595; AUROC 1 month 0.971, 1 year 0.928, 2 year 0.881, 5 year 0.801
  - Observation: Accuracy is better for advanced economies and for shorter forecasting horizons.

### Quantile regression for DD-PD mapping
- Relationship and motivation:
  - Scatter plots of DD vs. log PD (example: oil and gas sector, United States) show lower DD → higher PD (less negative log PDs).
  - For a fixed DD, PD can vary widely because DD captures asset value, volatility, market value, and risk-free rate, while profitability, liquidity, and cyclical stage influence market value indirectly.
- Quantile regression approach:
  - Use quantile regression (Koenker and Basset, 1978) to differentiate PD response across risk regimes.
  - Estimated quantiles τ = (0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95, 0.99). Higher quantile lines (closer to zero horizontal line on log PD plot) correspond to riskier regimes.
  - General linear quantile regression specification with N observations:
    - logPD_i = α_0(τ) + α_1(τ) DD_i + X...T βββ(τ) + ε_i ≡ QR_τ(DD_i, X...); τ ∈ T, i = 1,...,N(3)
    - X... is set of cyclical indicators (economy- or sector-specific).
- Establishing baseline risk regime:
  - Baseline risk regime QR_* chosen by minimizing squared differences between QR prediction and observed logPD:
    - τ_* ≡ arg min_τ { Σ_{i=1}^N (QR_τ(DD_i, X...) − logPD_i)^2 }.(4)
  - Baseline regime can be estimated at firm, portfolio, sector, group of sectors, or economy-wide level.

### Scenario analysis with the DD-PD model
- Overview:
  - Granular DD-PD mapping allows scenario analysis at individual firm level; firms need not face identical shocks or migrate to same risk regimes.
  - Example application: greening scenarios where oil and gas firms migrate to higher risk regimes and renewable firms to lower risk regimes.
- Three types of scenario analyses:
  - Shocks in the baseline regime (firm remains in current risk regime)
  - Shocks in the stress regime (firm migrates to higher risk regime)
  - Shocks in arbitrary risk regimes (migration to any specified quantile/regime)
- Common choices required for any scenario:
  - Values of four primary DD parameters: market value of firm, asset volatility of firm, risk-free rate, default barrier — set at firm, sector/industry, or economy-wide level.
  - Values of covariates X... used in QR estimation (typically apply equally across firms).
  - Choice of risk regime and firms’ risk regime migration; not necessary that all firms share same starting or ending regime.
- Use of empirical models or expert judgment:
  - Scenario values for DD parameters and covariates can come from empirical/quantitative models or from informed narratives and historical analogs; expert judgment guides scenario risk regime choice.

### Algorithms for scenario calculations
- Algorithm 2: Baseline risk regime analysis
  - Data: Observed pair (DD, PD), shock parameters, estimated QR_{τ_*}(DD, X...) baseline mapping.
  - Result: Return post-shock (DD_s, PD_s).
  - Steps:
    1. Compute post-shock DD_s using equations (1) and (2) and shock parameters.
    2. Use baseline QR to compute model fitted PDs for pre- and post-shock DD values, i.e. logPD^q and logPD^q_s.
    3. Compute logPD_s = logPD × logPD^q_s / logPD^q ̇;
- Algorithm 3: Stress analysis
  - Data: Observed pair (DD, PD), shock parameters, estimated QR_{τ_*}(DD, X...) baseline mapping.
  - Result: Return post-stress (DD_s, PD_s).
  - Steps:
    1. Compute post-shock DD_s using equations (1) and (2) and shock parameters.
    2. Use baseline QR to compute model fitted PDs for pre-shock DD values, i.e. logPD^q and use high risk quantile regression (95 percentile) to compute model fitted logPD^q_s.
    3. Compute logPD_s = logPD × logPD^q_s / logPD^q ̇;
- Algorithm 4: Arbitrary risk regime analysis
  - Algorithm 3 also applies when specifying an arbitrary target quantile (e.g., 25 percent) instead of the 95 percentile.
  - Steps:
    1. Compute post-shock DD_s using equations (1) and (2) and shock parameters.
    2. Use baseline QR to compute model fitted PDs pre-shock (logPD^q) and the arbitrary risk regime QR desired (e.g., 25 percent) to compute fitted logPD^q_a.
    3. Compute logPD_s = logPD × logPD^q_a / logPD^q ̇;
- Practical note:
  - Algorithms 2 and 3 are applied in subsequent sections to perform baseline risk regime and stress analyses of corporate default risk in the United States under different market correction scenarios.

*Italic: Source — wpiea2021143-print-pdf (excerpt provided).*

### 5. Scenario Analysis: Corporate Sector Risk in the United States

### 5. Scenario Analysis: Corporate Sector Risk in the United States

### Scenarios
- Three stock market and volatility shock scenarios considered using the DD-PD mapping:
  - Scenario 1: a broad market correction of and an increase in asset volatility by 10 percent,
  - Scenario 2: a broad market correction of and an increase in asset volatility by 20 percent, and
  - Scenario 3: a broad market correction of and an increase in asset volatility by 30 percent.
- Stress analysis: stress regime set equal to the 95th quantile regression with current date set to end-December 2020.
- Note: the price correction in Scenario 3 is similar to the stock market correction observed after the bursting of the 2000 dotcom bubble in the United States.

### Data and estimation
- Sample coverage and sources:
  - Over 4600 publicly listed firms (including dead and currently listed firms), sample period August 1985–December 2020.
  - Monthly market value, short-term and long-term liabilities from Thomson Reuters.
  - 1-year probability of default time series from Credit Research Initiative, National University of Singapore.
  - 1-year risk free rate = yield of the 1-year U.S. Treasury bill.
- Asset value and volatility estimation:
  - Calculated using observations in the past 60-month period.
  - If less than 60 observations, used all past observations provided at least 12 monthly data points existed.
  - Resulting asset volatilities correspond to medium-term asset volatilities.
- Liabilities handling and dataset size:
  - Short-term and long-term liabilities reported quarterly; assumed constant between quarterly reports.
  - Original raw dataset ≈ 9 million observations; processed usable observations ≈ 700,000 observations.
- Cyclical covariates used in quantile regressions (all seasonally adjusted):
  - Conference Board Leading Economic Index,
  - Capacity Utilization Index,
  - Consumer Confidence Index,
  - ISM Non-Manufacturers’ Survey Index,
  - ISM Purchasing Managers’ Index,
  - unemployment rate.
- PD-DD mapping estimation:
  - Estimated individually for 65 industrial subsectors, aggregated into nine industrial sectors.
  - Quantiles estimated: 5, 10, 25, 50, 75, 90, 95, and 99.
- Industrial sectors and subsector counts (excerpted):
  - Basic materials (88 firms)
  - Communications (174 firms)
  - Consumer, Cyclical (362 firms)
  - Consumer, non-cyclical (661 firms)
  - Energy (184 firms)
  - Financial (805 firms)
  - Industrial (369 firms)
  - Technology (242 firms)
  - Utilities (68 firms)

### Results — Asset volatility and leverage
- Historical patterns:
  - Medium-term volatilities high after 1997 Asian financial crisis and 1998 Russian sovereign default, then trended downwards.
  - Spike in PD during 2008 global financial crisis driven mainly by drastic stock market correction rather than increased asset volatility.
  - Quantitative easing pressured medium-term volatility downwards until 2013; since then asset volatility has been rising in line with the 1-year PD.
- Sectoral end-December 2020 observations:
  - Most volatile sectors: communications, consumer non-cyclical, energy, and technology.
  - Leverage measured as default barrier (short-term debt + one half long-term debt) divided by asset value.
  - Non-cyclical and technology sectors: high volatility partly offset by lower leverage ratios.

### Results — Baseline risk regimes (end-December 2020)
- Baseline risk regimes identified via quantile regression; selected industry group examples with quantiles:
  - Diversified Financial Services: 25
  - Investment Companies: 25
  - Real Estate: 25
  - Savings & Loans: 25
  - Auto Manufacturers: 50
  - Home Builders: 50
  - Textiles: 50
  - Agriculture: 50
  - Household Products/Wares: 50
  - Energy-Alternate Sources: 50
  - Banks: 50
  - Insurance: 50
  - REITS: 50
  - Environmental Control: 50
  - Semiconductors: 50
  - Software: 50
  - Chemicals: 75
  - Forest Products & Paper: 75
  - Iron/Steel: 75
  - Internet: 75
  - Telecommunications: 75
  - Distribution/Wholesale: 75
  - Home Furnishings: 75
  - Housewares: 75
  - Building Materials: 75
  - Electrical Components & Equipment: 75
  - Electronics: 75
  - Machinery-Construction & Mining: 75
  - Metal Fabricate/Hardware: 75
  - Packaging & Containers: 75
  - Transportation: 75
  - Trucking & Leasing: 75
  - Computers: 75
  - Media: 95
  - Retail: 95
  - Toys/Games/Hobbies: 95
  - Healthcare-Services: 95
  - Pharmaceuticals: 95
  - Oil & Gas Services: 95
  - Pipelines: 95
  - Office/Business Equipment: 95
  - Advertising: 99
  - Apparel: 99
  - Office Furnishings: 99
  - Beverages: 99
  - Coal: 99
  - Airlines: 90
  - Auto Parts & Equipment: 90
  - Entertainment: 90
  - Leisure Time: 90
  - Lodging: 90
  - Biotechnology: 90
  - Commercial Services: 90
  - Mining: 90
  - Aerospace/Defense: 90
  - Engineering & Construction: 90
  - Hand/Machine Tools: 90
  - Machinery-Diversified: 90
  - Miscellaneous Manufacturing: 90
  - Water: 90
- Interpretation:
  - Construction, real estate, and financial services groups generally in normal and low risk regimes (quantiles ≤ 50).
  - 3 of 4 industry sectors are in high risk regimes; particularly apparel, office equipment, and energy (except green energy companies).

### Results — Probabilities of default (PD) analysis
- Liability-weighted PDs and implied S&P ratings aggregated into nine industrial sectors at end-December 2020 and for each scenario and regime (PDs in bps; implied ratings per CRI):
  - Basic materials: Dec. 2020 = 22; Scenario 1 = 30; Scenario 2 = 48; Scenario 3 = 82; Stress Scenario 1 = 117; Stress Scenario 2 = 161
    - Implied ratings: baseline and stress bands shown in table.
  - Communications: Dec. 2020 = 27; Scenario 1 = 34; Scenario 2 = 45; Scenario 3 = 59; Stress Scenario 1 = 54; Stress Scenario 2 = 71; Stress Scenario 3 = 90
  - Consumer, cyclical: Dec. 2020 = 63; Scenario 1 = 77; Scenario 2 = 98; Scenario 3 = 124; Stress Scenario 1 = 137; Stress Scenario 2 = 170; Stress Scenario 3 = 206
  - Consumer, non-cyclical: Dec. 2020 = 29; Scenario 1 = 37; Scenario 2 = 51; Scenario 3 = 71; Stress Scenario 1 = 58; Stress Scenario 2 = 93; Stress Scenario 3 = 148
  - Energy: Dec. 2020 = 152; Scenario 1 = 181; Scenario 2 = 226; Scenario 3 = 279; Stress Scenario 1 = 120; Stress Scenario 2 = 151; Stress Scenario 3 = 187
  - Financial: Dec. 2020 = 23; Scenario 1 = 31; Scenario 2 = 42; Scenario 3 = 58; Stress Scenario 1 = 17; Stress Scenario 2 = 82; Stress Scenario 3 = 292; Stress Scenario 3 value noted in table
  - Industrial: Dec. 2020 = 25; Scenario 1 = 36; Scenario 2 = 59; Scenario 3 = 93; Stress Scenario 1 = 75; Stress Scenario 2 = 116; Stress Scenario 3 = 168
  - Technology: Dec. 2020 = 17; Scenario 1 = 23; Scenario 2 = 36; Scenario 3 = 55; Stress Scenario 1 = 89; Stress Scenario 2 = 125; Stress Scenario 3 = 169
  - Utilities: Dec. 2020 = 23; Scenario 1 = 36; Scenario 2 = 64; Scenario 3 = 105; Stress Scenario 1 = 112; Stress Scenario 2 = 172; Stress Scenario 3 = 247
- Key observations:
  - In baseline regimes, prompt implied rating downgrades of one to two notches; in stress regimes, up to five notches.
  - Implied rating bands can be wide and mask substantial changes in creditworthiness (example: stress regime weighted implied rating stays flat at B+ despite a 100 bps PD difference between worst and least severe shocks).
  - Sectoral PD distributions generally shift right and develop fatter right tails under shocks; exception: energy sector under Scenario 1 where PD distribution shifts left and narrows due to migration from baseline 99th quantile to the imposed 95th stressed quantile.
  - Technology and financial sectors are particularly sensitive to shocks despite technology’s low leverage and financials’ low asset volatility.
  - Cross-sector inferences are limited by differing DD-PD mappings across sectors and subsectors.

### Results — Economic capital in corporate loan portfolios
- Baseline median 1-year PD at end-December 2020 = 10 bps (equivalent to S&P BBB-rating).
  - Global historical 1-year transition rate associated with this rating = 0.16 percent.
- Main systemic concern: need for lending institutions to increase provisions and economic capital buffers rather than defaults per se.
  - Higher PDs could impair corporations’ access to credit; lenders may curtail lending to meet regulatory capital requirements; institutional investors may reduce exposure if ratings fall below investment grade; hedging costs could rise.
- Capital requirement formula used (Basel II ASRF IRB function without maturity adjustment):
  - K = LGD × Φ( √(1−R) × Φ−1(PD) + √(R/(1−R)) × Φ−1(0.999) ) − PD × LGD
  - Parameters: LGD = 40 percent; R = 30 percent.
- Median economic capital requirements reported as multiples of December 2020 values (Table values):
  - Basic materials: Baseline Scenario 1 = 1.21; Scenario 2 = 1.56; Scenario 3 = 1.88; Stress Scenario 1 = 1.84; Stress Scenario 2 = 2.13; Stress Scenario 3 = 2.32
  - Communications: Baseline 1.17, 1.45, 1.70; Stress 1.68, 1.90, 2.03
  - Consumer, cyclical: Baseline 1.14, 1.35, 1.52; Stress 1.23, 1.42, 1.56
  - Consumer, non-cyclical: Baseline 1.15, 1.41, 1.64; Stress 1.37, 1.59, 1.78
  - Energy: Baseline 1.05, 1.13, 1.19; Stress 1.01, 1.02, 1.06
  - Financial: Baseline 1.29, 1.78, 2.24; Stress 3.17, 3.22, 3.23
  - Industrial: Baseline 1.21, 1.57, 1.92; Stress 1.82, 2.12, 2.31
  - Technology: Baseline 1.28, 1.87, 2.45; Stress 3.23, 3.50, 3.80
  - Utilities: Baseline 1.28, 1.73, 2.14; Stress 2.07, 2.41, 2.65
- Aggregate impacts:
  - In baseline risk regimes, price shocks of 10 percent, 20 percent, and 30 percent cause median capital requirements to increase on average by 20 percent, 54 percent, and 85 percent respectively.
  - In stress regime, increases are 94 percent, 114 percent, and 130 percent respectively.
  - Energy sector exposures (already in very high risk/stressed regimes) do not require substantial capital increases under these scenarios.

### Conclusions and policy-relevant implications
- DD-PD mapping links equity-based distance-to-default to real-world PDs via quantile regression; each quantile regression line can be associated with a risk regime.
- Uses of the mapping:
  - Identify current risk regime of individual firms, industry groups, and sectors from a bottom-up perspective.
  - Perform what-if scenario analysis by simulating shocks to DD determinants under baseline or stressed risk regimes.
  - Applications include managing corporate credit portfolio risk and informing regulatory stress testing.
- Empirical application to U.S. corporate sector:
  - A sustained price correction of 10 percent could cause PDs to rise 30 to 70 percent above end-December 2020 levels; most affected sectors: energy, financial, and technology.
  - Under stress analysis, PDs could be one order of magnitude higher in financial and technology sectors.
- Additional potential uses (suggested, not explored in detail):
  - Analyze effects of changes in firms’ liability structures on PD and optimal capital structure.
  - Examine impact of monetary policy on firms’ default risk by assessing relationship between equity prices and the risk-free rate and applying the DD-PD mapping.
  - Model potential migration across risk regimes for firms or sectors under secular or cyclical changes in exogenous covariates to guide credit portfolio composition and long-term cost-of-capital projections (e.g., climate change investment scenarios).

*Source: wpiea2021143-print-pdf - 5. Scenario Analysis: Corporate Sector Risk in the United States*

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*References list from wpiea2021143-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021143-print-pdf.pdf_