Credit Risk Models

Why Credit Risk Models?

Credit risk — the possibility that a borrower will fail to meet obligations — is one of the oldest and most significant risks in finance. Unlike market risk, which involves price fluctuations of liquid assets, credit risk involves discrete events (default) with potentially large losses and complex recovery processes.

Modern credit risk modeling became essential after the 1990s, driven by Basel regulations, the growth of credit derivatives, and major defaults like Enron and Lehman Brothers. These models are crucial for loan pricing, portfolio management, regulatory capital, and trading credit derivatives.

Types of Credit Risk

1. Default Risk

The probability that a borrower will fail to make required payments.

2. Migration Risk

The risk that a borrower's creditworthiness deteriorates (rating downgrade) without actual default.

3. Recovery Risk

Uncertainty in the amount recovered if default occurs.

4. Concentration Risk

Risk arising from lack of diversification in the credit portfolio.

5. Counterparty Risk

Credit risk in derivatives and other bilateral contracts.

Structural Models

Merton Model (1974)

Framework: Default occurs when firm value falls below debt level.

Setup:

Firm value: $V_t = V_0 e^{(r-\frac{\sigma^2}{2})t + \sigma W_t}$
Debt: Face value $D$ maturing at time $T$
Default: Occurs if $V_T < D$

Equity Value:

E_0 = V_0 N(d_1) - De^{-rT} N(d_2)

where:

d_1 = \frac{\ln(V_0/D) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}

Default Probability:

\mathbb{P}(\text{default}) = N(-d_2)

Extensions of Merton Model

1. First-Passage Models

Default can occur anytime when firm value hits barrier

B < D

\tau = \inf\{t \geq 0 : V_t \leq B\}

Default Probability:

\mathbb{P}(\tau \leq T) = N\left(\frac{\ln(B/V_0) + (r-\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\right) + \left(\frac{B}{V_0}\right)^{2r/\sigma^2-1} N\left(\frac{\ln(B/V_0) - (r-\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\right)

2. Jump-Diffusion Extensions

Firm value follows jump-diffusion process:

dV_t = rV_t dt + \sigma V_t dW_t + V_{t-}(e^J - 1)dN_t

This captures sudden deteriorations in firm fundamentals.

3. Stochastic Barrier Models

The default barrier itself is stochastic:

dB_t = \mu_B B_t dt + \sigma_B B_t dW_t^B

Calibration Challenges

Unobservable firm value: Must be inferred from equity prices
Volatility estimation: Equity volatility ≠ asset volatility
Parameter stability: Model parameters change over time
Low default frequencies: Difficult to validate empirically

Reduced-Form Models

Hazard Rate Approach

Framework: Model the intensity (hazard rate) of default directly, without modeling firm value.

Default Time:

\tau

has intensity process

\lambda_t

Survival Probability:

\mathbb{P}(\tau > t) = e^{-\int_0^t \lambda_s ds}

Default Probability:

\mathbb{P}(\tau \leq t) = 1 - e^{-\int_0^t \lambda_s ds}

Constant Hazard Rate

If $\lambda_t = \lambda$ (constant):

Survival function: $S(t) = e^{-\lambda t}$
Default density: $f(t) = \lambda e^{-\lambda t}$
Exponential distribution for default times

Stochastic Hazard Rates

Cox-Ingersoll-Ross (CIR) Process

d\lambda_t = \kappa(\theta - \lambda_t)dt + \sigma\sqrt{\lambda_t} dW_t

Features:

Mean reversion: High default risk tends to decrease
Non-negative: Square-root diffusion keeps $\lambda_t \geq 0$
Affine structure: Analytical bond pricing formulas

Vasicek Process

d\lambda_t = \kappa(\theta - \lambda_t)dt + \sigma dW_t

Features:

Gaussian: Can become negative (less realistic)
Analytical tractability: Closed-form survival probabilities

Correlated Defaults

For portfolio models, hazard rates can be correlated:

d\lambda_{i,t} = \kappa_i(\theta_i - \lambda_{i,t})dt + \sigma_i dW_{i,t}

where $dW_{i,t} dW_{j,t} = \rho_{ij} dt$ .

Credit Derivatives Pricing

Credit Default Swaps (CDS)

Structure: Insurance against default of reference entity. Premium Leg: Periodic payments of spread

s

times notional Protection Leg:

(1-R) \times \text{Notional}

if default occurs

Pricing Equation:

\text{Spread} \times \text{PV of Premium Leg} = (1-R) \times \text{Default Probability} \times \text{Discount Factor}

Risky Bond Pricing

Bond Price:

P(0,T) = \mathbb{E}\left[e^{-\int_0^{\tau \wedge T} r_s ds} \mathbf{1}_{\{\tau > T\}} + e^{-\int_0^\tau r_s ds} R \mathbf{1}_{\{\tau \leq T\}}\right]

Components:

Survival: Full repayment if no default
Default: Recovery $R$ if default occurs

Credit Spread

Credit Spread:

s = y_{\text{risky}} - y_{\text{risk-free}}

Relationship to Default Probability:

s \approx \lambda (1-R)

for small spreads, where $\lambda$ is the hazard rate.

Portfolio Credit Risk Models

Gaussian Copula Model

Framework: Joint defaults driven by correlated normal variables.

Setup:

Y_i = \sqrt{\rho} X + \sqrt{1-\rho} Z_i

where $X, Z_i \sim \mathcal{N}(0,1)$ independent.

Default: Firm

i

defaults if

Y_i < \Phi^{-1}(PD_i)

Portfolio Loss:

L = \sum_{i=1}^N (1-R_i) EAD_i \mathbf{1}_{\{\text{default}_i\}}

One-Factor Model

Vasicek (1987): Industry standard for regulatory capital.

Asset Value:

A_i = \sqrt{\rho} X + \sqrt{1-\rho} \varepsilon_i

Default: If

A_i < \Phi^{-1}(PD_i)

Conditional Default Probability:

\mathbb{P}(\text{default}_i | X = x) = \Phi\left(\frac{\Phi^{-1}(PD_i) - \sqrt{\rho} x}{\sqrt{1-\rho}}\right)

Multi-Factor Models

Extend to multiple systematic risk factors:

A_i = \sum_{k=1}^K \sqrt{\rho_{i,k}} X_k + \sqrt{1-\sum_{k=1}^K \rho_{i,k}} \varepsilon_i

Applications:

Industry factors: Technology, financial, energy
Geographic factors: US, Europe, Asia
Macroeconomic factors: GDP, interest rates

Copula Models

Definition

A copula

C(u_1, \ldots, u_n)

links univariate marginal distributions to form a multivariate distribution:

F(x_1, \ldots, x_n) = C(F_1(x_1), \ldots, F_n(x_n))

Popular Credit Copulas

1. Gaussian Copula

C^{\text{Gauss}}(u_1, \ldots, u_n) = \Phi_n(\Phi^{-1}(u_1), \ldots, \Phi^{-1}(u_n); \Sigma)

where $\Phi_n$ is the multivariate normal CDF with correlation matrix $\Sigma$ .

2. t-Copula

C^t(u_1, \ldots, u_n) = t_{\nu,\Sigma}(t_\nu^{-1}(u_1), \ldots, t_\nu^{-1}(u_n))

Features:

Heavy tails: More realistic extreme dependence
Tail dependence: Non-zero probability of joint extreme events

3. Archimedean Copulas

Generated by function $\varphi$ :

C(u_1, \ldots, u_n) = \varphi^{-1}(\varphi(u_1) + \cdots + \varphi(u_n))

Examples:

Clayton: Lower tail dependence
Gumbel: Upper tail dependence
Frank: Symmetric dependence

Calibration

Maximum Likelihood:

\max_{\theta} \sum_{t=1}^T \ln c(u_{1,t}, \ldots, u_{n,t}; \theta)

where $c$ is the copula density.

Credit Risk Metrics

Value at Risk (VaR)

Credit VaR: Maximum expected loss due to credit events at confidence level

\alpha

\text{CVaR}_\alpha = F_L^{-1}(\alpha) - \mathbb{E}[L]

Expected Shortfall (ES)

\text{ES}_\alpha = \mathbb{E}[L | L \geq \text{VaR}_\alpha]

Economic Capital

Definition: Capital needed to absorb losses at high confidence level (e.g., 99.9%) over one year.

Basel III:

\text{EC} = K \times \text{EAD}

where

K

is the capital ratio.

Unexpected Loss

\text{UL} = \sqrt{\text{Var}(L)}

Components:

\text{UL}^2 = \sum_{i=1}^N \text{UL}_i^2 + 2\sum_{i<j} \rho_{ij} \text{UL}_i \text{UL}_j

Monte Carlo Simulation

Basic Algorithm

Generate systematic factors: $X \sim \mathcal{N}(0,1)$
Generate idiosyncratic shocks: $Z_i \sim \mathcal{N}(0,1)$
Compute asset values: $A_i = \sqrt{\rho} X + \sqrt{1-\rho} Z_i$
Determine defaults: If $A_i < \Phi^{-1}(PD_i)$
Calculate portfolio loss: $L = \sum_{\text{defaults}} (1-R_i) EAD_i$
Repeat for many scenarios

Variance Reduction

Importance Sampling

Sample from modified distribution to increase default probability in tail scenarios.

Stratified Sampling

Divide systematic factor space into strata and sample within each.

Antithetic Variables

Use pairs $(X, -X)$ to reduce variance.

Stress Testing

Regulatory Stress Tests

CCAR (US): Comprehensive Capital Analysis and Review EBA (EU): European Banking Authority stress tests

Scenarios:

Baseline: Expected economic conditions
Adverse: Recession scenario
Severely adverse: Financial crisis scenario

Methodologies

Historical scenarios: Replicate past crises
Hypothetical scenarios: Expert judgment on tail risks
Monte Carlo: Simulate many possible scenarios

Machine Learning in Credit Risk

Credit Scoring

Traditional: Logistic regression, linear discriminant analysis Modern: Random forests, gradient boosting, neural networks

Features

Financial ratios: Leverage, profitability, liquidity
Market data: Stock prices, volatility, CDS spreads
Alternative data: Social media, transaction data
Macroeconomic: GDP, unemployment, interest rates

Model Validation

Out-of-sample testing: Performance on unseen data
Backtesting: Historical performance evaluation
Stress testing: Performance under adverse conditions
Model interpretability: Regulatory requirements

Regulatory Framework

Basel III

Key Components:

Minimum capital ratios
Risk-weighted assets calculation
Leverage ratio
Liquidity requirements

Internal Ratings-Based (IRB) Approach

Banks develop own models for:

PD: Probability of default
LGD: Loss given default
EAD: Exposure at default

Capital Requirement:

K = (LGD \times N(\sqrt{\frac{1}{1-R}}N^{-1}(PD) + \sqrt{\frac{R}{1-R}}N^{-1}(0.999)) - PD \times LGD) \times M

Connection to Other Topics

Credit risk models integrate multiple quantitative areas:

Built on Stochastic Differential Equations for firm value
Use Jump-Diffusion Processes for sudden deteriorations
Apply Martingale pricing for derivatives
Connect to Interest Rate Models for discounting
Foundation for fixed-income derivatives
Link to survival analysis in statistics
Enable sophisticated risk management and regulatory compliance