Credit Risk Modeling

Credit risk — the risk that a borrower will fail to fulfill obligations — is the primary risk for banks (60–70% RWA for commercial banks). Its management is regulated by the Basel Accords. Mathematical models of credit risk are critically important for pricing corporate bonds, CDS, calculation of loss reserves (IFRS 9, CECL). There are two major families: structural models (based on asset value) and reduced-form models (based on default intensity). Their comparison and combination is the main question of modern credit analytics.

Structural Models (Merton, 1974)

Merton's idea. The company's assets $V_t$ are modeled as a geometric Brownian motion: $ dV = μ·V·dt + σ·V·dW. $

The company's debt $D$ is repaid at time $T$. Default occurs if $V_T < D$ (assets are insufficient for the debt).

Option approach. Equity (stock value): $ E_T = \max(V_T − D, 0). $

This is a call option on assets with strike $D$! Applying the Black-Scholes formula: $ E_0 = V_0·N(d_1) − D·e^{−r·T}·N(d_2), $ where $ d_1 = \frac{\ln(V_0/D) + (r + σ²/2)·T}{σ·\sqrt{T}}, \quad d_2 = d_1 − σ·\sqrt{T}. $

Debt: $D_0 = V_0 − E_0$ — value for bondholders.

Default probability (real-world): $P(V_T < D) = N(−DD)$, where DD = (ln(V_0/D) + (μ − σ²/2)·T)/(σ·\sqrt{T}) — distance-to-default. The higher the DD, the further the company is from default.

Risk-neutral default probability: $N(−d_2)$ with $μ \to r$.

Merton Model Extensions

Black-Cox (1976): Default can occur at any moment upon first hitting the barrier (not only at $T$). More realistic.

KMV / Moody's Credit Monitor. Commercial implementation of Merton: $V_0$ and $σ$ are estimated from $E_0$ (market capitalization) and $σ_E$ (stock volatility) using Newton's method. DD is converted into Expected Default Frequency (EDF) via empirical calibration. Widely used in industry.

Reduced-Form Models (Jarrow-Turnbull 1995, Duffie-Singleton 1999)

Idea. We do not model assets, but directly — default intensity $h(t)$ (hazard rate). Default is the first jump time of a Poisson process with intensity $h(t)$.

Survival probability: $P(τ > t) = \exp(−\int_0^t h(s) ds)$.

Pricing a corporate bond (zero-coupon, recovery $R$ in case of default): $ B(0, T) = E^Q[\exp(−\int_0^T r_s ds)·(1·1_{τ>T} + R·1_{τ≤T})] $ $ ≈ \exp(−r·T) · [(1 − R)·\exp(−h·T) + R]. $

With constant $h$: the spread over the risk-free rate is approximately $h·(1 − R)$.

CDS Spread

CDS (Credit Default Swap). Default insurance: the buyer pays a regular spread $s$, and receives $(1 − R)·N$ (notional × loss given default) upon default.

No-arbitrage condition: $s ≈ h·(1 − R)$. When calculating CDS spread from market quotes (bootstrapping), risk-neutral default intensity is recovered.

Example: 5-year CDS on a corporation trades at 200 bps = 2%. With $R = 40%$: $h ≈ 2%/(1 − 0.4) = 3.33%$ annual risk-neutral default probability.

Credit Ratings and Migration Matrix

Rating agencies (S&P, Moody's, Fitch) assign ratings from AAA (low risk) to D (default). Historical default tables:

• AAA: $P(\text{default over 5 years}) = 0.05%$.
• AA: 0.20%.
• A: 0.50%.
• BBB: 2.5%.
• BB: 12%.
• B: 25%.
• CCC: 50%.

Migration matrix (rating transition matrix). $P_{ij}$ = probability of transitioning from rating $i$ to $j$ in one year. Markov chain (simplification of reality — actually there is a "memory effect"). Long-term default probabilities from $P^n$.

CreditMetrics (J.P. Morgan, 1997). Uses ratings + migration matrix + correlations to calculate credit portfolio VaR.

Numerical Example: Merton for a Company

$V_0 = 100$, $D = 80$, $σ = 0.25$, $μ = 0.06$, $r = 0.03$, $T = 1$.

$ DD = \frac{\ln(100/80) + (0.06 − 0.0625/2)·1}{0.25·1} = \frac{0.2231 + 0.0288}{0.25} = 1.008. $ $P(\text{default real-world}) = N(−1.008) = 0.157 \rightarrow 15.7%$ annual.

$P(\text{default risk-neutral}) = N(−d_2)$, $d_2 = \frac{\ln(100/80) + (0.03 − 0.03125)·1}{0.25} − 0 = 0.886$. $P_Q = N(−0.886) = 0.188 \rightarrow 18.8%$.

With $R = 0.4$: fair spread for a 5-year bond: $ s ≈ −(1/T)·\ln(1 − P_{\text{default}} · (1 − R)) ≈ 0.188 · 0.6 / 1 = 11.3% \quad (\text{roughly}). $ More precisely via NHPP calibration: $h ≈ 0.21$ (annual intensity). 5-year spread ≈ $h·(1−R) = 12.5%$.

With $σ = 0.35$ (higher volatility): $DD = (0.2231 + (0.06 − 0.0613)·1)/0.35 = 0.633$. $P_{def} = N(−0.633) = 0.263 \rightarrow 26.3%$. The spread rises sharply. Lesson: asset volatility is the main driver of credit risk.

IFRS 9 and Expected Credit Loss (ECL)

Since 2018, banks have switched from incurred loss (IAS 39) to expected loss (IFRS 9). Reserve = probability-weighted sum of losses:

ECL = PD · LGD · EAD, where:

• PD (Probability of Default): probabilistic over horizon (12 months or lifetime).
• LGD (Loss Given Default): $1 − R$, usually 35–50% for unsecured loans.
• EAD (Exposure at Default): loan balance at the moment of default.

Three stages: Stage 1 (12-month ECL), Stage 2 (lifetime ECL, significant increase in credit risk), Stage 3 (lifetime ECL, defaulted).

Real Applications

• Corporate lending. Sberbank, Alfa: internal PD models (logistic regression on financial indicators + behavioral data), LGD, EAD. RWA calculation according to Basel IRB.
• CDS market. $9$ trillion notional outstanding. ICE Clear Credit, LCH SwapClear as central counterparties.
• Crisis 2008. AIG sold $400$ billion CDS on subprime MBS, without enough capital to cover defaults — government bail-out $182$ billion.
• Moody's Credit Monitor / KMV. Used by > 200 banks for real-time monitoring of clients' PD.
• Credit cards, consumer loans. FICO score (US), Equifax (UK), NBKI (Russia) — simplified reduced-form models for retail lending.

Assignment. A company with $V_0 = 100$, $D = 80$, $σ = 0.25$, $μ = 0.06$, $r = 0.03$, $T = 1$. (a) Compute distance-to-default and default probability via Merton (real-world and risk-neutral). (b) If recovery rate $R = 0.4$, what is the fair spread of a 5-year corporate bond? (c) Using Monte Carlo simulation (10,000 paths for $V_t$ via GBM with 100 steps per year) compute $P(\min_{t ≤ 1} V_t < 80)$ — first-passage probability. Compare with standard Merton (default only at $T$). (d) Sensitivity: how does the spread change for $σ ∈ {0.15, 0.25, 0.35, 0.50}$? Plot the graph.