credit-scoring-models

Credit scoring models — Altman Z-score, Merton, scorecards.

When to Activate

• Assessing the creditworthiness of a corporate borrower or counterparty
• Building or validating a credit scorecard for lending decisions
• Estimating probability of default (PD) for a portfolio
• Evaluating a structural model approach to credit risk
• Analyzing rating migration and transition probabilities
• Calibrating internal rating systems for regulatory capital (IRB approach)
• Back-testing model performance against realized defaults

Core Concepts

Altman Z-Score

Edward Altman's discriminant analysis model predicts corporate bankruptcy using five financial ratios. Originally developed for publicly traded manufacturers (1968).

Original Z-Score (public manufacturing):

Z = 1.2 * X1 + 1.4 * X2 + 3.3 * X3 + 0.6 * X4 + 1.0 * X5

X1 = Working Capital / Total Assets          (liquidity)
X2 = Retained Earnings / Total Assets        (cumulative profitability)
X3 = EBIT / Total Assets                     (operating efficiency)
X4 = Market Value Equity / Book Value Debt   (solvency)
X5 = Sales / Total Assets                    (asset turnover)

• Z > 2.99 — Safe zone (low default probability)
• 1.81 < Z < 2.99 — Grey zone (caution)
• Z < 1.81 — Distress zone (high default probability)

Z'-Score (private firms): Replaces market value of equity with book value. Revised coefficients and cutoffs (Z' < 1.23 = distress).

Z''-Score (non-manufacturing / emerging markets): Drops X5 (Sales/Total Assets) to remove industry bias from asset turnover. Suitable for service firms and emerging market companies.

• Apply the correct variant based on the entity type
• Z-score is a point-in-time indicator — supplement with trend analysis over 3-5 years
• Does not capture industry-specific risk factors or qualitative considerations

Merton Structural Model

Based on the Black-Scholes option pricing framework. Equity is modeled as a call option on the firm's assets with a strike price equal to the face value of debt.

• Asset value (V): Unobservable — inferred from equity market value and equity volatility using iterative methods
• Asset volatility (sigma_V): Also unobservable — estimated simultaneously with V
• Default point: Firm defaults when asset value falls below the debt obligation at maturity
• Distance to Default (DD): Number of standard deviations the asset value is above the default point

  DD = (ln(V/D) + (mu - 0.5 * sigma_V^2) * T) / (sigma_V * sqrt(T))
  

• PD from DD: Map DD to PD using the normal distribution (theoretical) or empirical mapping (Moody's KMV / EDF approach uses historical default frequency for each DD bucket)

Strengths: Market-based, forward-looking, continuous updating. Weaknesses: Requires liquid equity market, assumes single debt maturity, sensitive to equity volatility estimation.

Credit Scorecards

Statistical models (typically logistic regression) that assign points to borrower characteristics to produce a credit score.

Development process:

1. Data collection: Gather historical loan-level data with default outcomes (12-month observation window typical)
2. Variable selection: Financial ratios, behavioral data, industry, age of firm, management quality proxies
3. Weight of Evidence (WoE) transformation: Bin continuous variables; calculate WoE = ln(% of goods / % of bads) for each bin
4. Information Value (IV): Measures predictive power of each variable. IV > 0.3 = strong; 0.1-0.3 = medium; < 0.1 = weak
5. Logistic regression: Fit model using selected WoE-transformed variables
6. Scaling: Convert log-odds to a score. Common convention: Score = Offset + Factor * ln(odds), where Factor = PDO / ln(2), PDO = points to double the odds
7. Scorecard format: Each attribute level gets a partial score; total score maps to PD

Probability of Default (PD) Estimation

• Through-the-cycle (TTC): Long-run average PD over a full economic cycle — used for regulatory capital
• Point-in-time (PIT): Current PD reflecting prevailing economic conditions — used for IFRS 9 / CECL provisioning
• PD calibration: Ensure model-predicted PDs align with observed default rates. Central tendency adjustment to match long-run average
• Low-default portfolios: Where defaults are rare (e.g., investment-grade corporates, sovereigns), use external data, expert judgment, or Bayesian techniques to estimate PD

Rating Migration Matrices

Transition matrices show the probability of moving from one rating grade to another over a defined horizon (typically one year).

From \ To    AAA    AA     A      BBB    BB     B     CCC/D
AAA          90.0   8.5    1.0    0.3    0.1    0.0   0.1
AA           1.0    88.0   8.5    1.5    0.5    0.3   0.2
A            0.1    2.0    87.0   7.5    2.0    0.8   0.6
BBB          0.0    0.3    4.0    84.0   7.0    3.0   1.7
BB           0.0    0.1    0.5    5.0    78.0   10.0  6.4

• Upgrade/downgrade ratios: Track the health of a portfolio over time
• Absorbing state: Default is absorbing — once an entity defaults, it cannot migrate back
• Multi-year PDs: Derived by raising the one-year transition matrix to the power of n

Model Validation

• Discriminatory power: AUROC (area under ROC curve), Gini coefficient (= 2 * AUROC - 1), Kolmogorov-Smirnov statistic
• Calibration accuracy: Binomial test, Hosmer-Lemeshow test, traffic light approach (compare predicted PD to observed default rate by grade)
• Stability: Population Stability Index (PSI) measures drift in score distributions over time. PSI > 0.25 signals significant shift
• Backtesting: Compare predicted defaults to actual defaults; analyze by segment, time period, and rating grade

Methodology

1. Define scope: Identify the portfolio segment, default definition (e.g., 90 days past due, bankruptcy), and observation period
2. Data preparation: Collect financial statements, market data, behavioral data. Clean and validate. Apply exclusions (e.g., newly formed entities, data errors)
3. Model development: Choose model type (scorecard, structural, hybrid). Develop candidate variables, test statistical significance, build the model
4. Calibration: Map model output to PD. Validate central tendency against long-run default rates. Adjust for economic cycle if TTC PD required
5. Validation: Test discriminatory power, calibration accuracy, and stability. Compare against benchmark models
6. Implementation: Integrate into credit approval workflow, pricing, and portfolio monitoring. Define override policy
7. Ongoing monitoring: Track model performance quarterly. Re-estimate or recalibrate when PSI exceeds thresholds or discriminatory power degrades

Templates

Altman Z-Score Calculation

Company: [Name]                     Year: [Year]

                                    Value       Ratio
Working Capital                     $12.5M
Total Assets                        $85.0M      X1 = 0.147
Retained Earnings                   $28.0M      X2 = 0.329
EBIT                                $10.2M      X3 = 0.120
Market Value of Equity              $45.0M
Book Value of Total Debt            $35.0M      X4 = 1.286
Sales                               $92.0M      X5 = 1.082

Z-Score = 1.2(0.147) + 1.4(0.329) + 3.3(0.120) + 0.6(1.286) + 1.0(1.082)
        = 0.176 + 0.461 + 0.396 + 0.772 + 1.082
        = 2.887

Assessment: Grey zone — monitor closely. Declining from 3.15 prior year.
Key concern: Working capital deterioration (X1 dropped from 0.210 to 0.147)

Scorecard Output Summary

Borrower: [Name]            Application Date: [Date]

Attribute                    Value           WoE Bin      Points
Debt/EBITDA                  3.2x            2.5-4.0x     +35
Interest Coverage            4.5x            3.0-5.0x     +28
Current Ratio                1.4x            1.2-1.6x     +22
Revenue Growth (3yr)         8%              5-10%        +18
Years in Business            12              10-20        +15
Industry Risk                Medium          B            +10
Management Quality           Strong          A            +20

Total Score: 148 + Base Score (200) = 348
Mapped PD: 0.85%
Internal Rating: BB+

Model Validation Dashboard

Metric                    Current Period   Prior Period   Threshold   Status
AUROC                     0.82             0.84           > 0.70      Pass
Gini Coefficient          0.64             0.68           > 0.40      Pass
KS Statistic              0.52             0.55           > 0.30      Pass
PSI (score distribution)  0.12             0.08           < 0.25      Pass
Hosmer-Lemeshow (p-value) 0.35             0.42           > 0.05      Pass
Predicted vs Actual DR    1.2% vs 1.4%    1.1% vs 1.0%  Within 20%  Pass

Quality Gate

• Correct Z-score variant applied for the entity type (public, private, non-manufacturing, emerging market)
• Merton model inputs validated: equity value, equity volatility, debt structure, risk-free rate
• Scorecard developed on representative data with adequate default observations
• Variable selection justified statistically (IV, p-values) and economically (business logic)
• PD calibration aligned with chosen philosophy (TTC or PIT) and long-run default rates
• Model validation completed: AUROC, Gini, KS, calibration tests all within thresholds
• Population Stability Index monitored; recalibration triggered if PSI > 0.25
• Rating migration matrix computed and compared to external benchmarks
• Model documentation meets regulatory standards (IRB, IFRS 9, CECL)
• Override rate tracked and within policy limits (typically < 10-15% of decisions)
• Independent model validation function has signed off