How does the Merton model work for measuring credit risk, and what does the structural diagram look like?

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AcadiFi · Accepted Answer

The Merton model is the foundational structural credit risk model tested on FRM Part II. It connects a firm's capital structure to option pricing theory, treating equity as a call option on the firm's assets with a strike price equal to the face value of debt. **Core Insight:** At debt maturity, if asset value (V) exceeds the debt face value (D), equityholders keep the residual: Equity = max(V - D, 0). If V < D, the firm defaults and equityholders receive nothing. This payoff is identical to a European call option on V with strike D. ```mermaid graph TD A[Firm Asset Value V follows GBM] --> B{At Maturity T: V vs D?} B -->|V > D: No Default| C[Equity = V - D] B -->|V < D: Default| D2[Equity = 0] C --> E[Debtholders receive D in full] D2 --> F[Debtholders receive V which is less than D] A --> G[Distance to Default DD] G --> H[DD = ln V/D + mu - 0.5 sigma squared times T divided by sigma times sqrt T] H --> I[PD = N of negative DD] style A fill:#c9a84c,color:#0a0a0f style B fill:#1e3a5f,color:#fff style C fill:#1a3c2a,color:#fff style D2 fill:#8b2020,color:#fff style I fill:#2d1b4e,color:#fff ``` **Model Assumptions:** 1. Firm asset value V follows geometric Brownian motion: dV = mu*V*dt + sigma*V*dW 2. The firm has a single zero-coupon debt with face value D maturing at time T 3. Default can only occur at maturity (not before) 4. Markets are frictionless (no taxes, no bankruptcy costs) **Worked Example:** Crestview Manufacturing has the following parameters: | Parameter | Value | |---|---| | Current asset value (V0) | $850 million | | Face value of debt (D) | $600 million | | Asset volatility (sigma) | 28% | | Risk-free rate (r) | 4.5% | | Debt maturity (T) | 3 years | | Expected asset return (mu) | 8.0% | **Step 1 — Calculate Distance to Default (DD):** DD = [ln(V0/D) + (mu - 0.5 * sigma^2) * T] / (sigma * sqrt(T)) DD = [ln(850/600) + (0.08 - 0.5 * 0.0784) * 3] / (0.28 * sqrt(3)) DD = [0.3483 + (0.0408) * 3] / (0.28 * 1.7321) DD = [0.3483 + 0.1224] / 0.4850 DD = 0.4707 / 0.4850 **DD = 0.9705** **Step 2 — Calculate Probability of Default:** PD = N(-DD) = N(-0.9705) Using the standard normal table: N(-0.97) = 0.1660 **PD = 16.60%** This means there is approximately a 16.6% probability that Crestview's assets will fall below $600 million at maturity, triggering default. **Step 3 — Value the Equity (using Black-Scholes):** d1 = [ln(850/600) + (0.045 + 0.5 * 0.0784) * 3] / (0.28 * 1.7321) d1 = [0.3483 + 0.2526] / 0.4850 = **1.2390** d2 = d1 - sigma * sqrt(T) = 1.2390 - 0.4850 = **0.7540** Equity = V0 * N(d1) - D * e^(-rT) * N(d2) Equity = 850 * N(1.239) - 600 * e^(-0.135) * N(0.754) Equity = 850 * 0.8923 - 600 * 0.8737 * 0.7746 Equity = 758.46 - 406.02 = **$352.4 million** **Model Limitations:** 1. Default only at maturity — real firms can default at any time (Black-Cox extends this) 2. Single debt class — real capital structures are complex 3. Asset value is unobservable — must be implied from equity price and volatility 4. Constant volatility assumption — real asset volatility varies with leverage **Exam Tip:** Know how to calculate DD and PD, and understand that the Merton model's PD corresponds to the risk-neutral probability (using r) for pricing, or the physical probability (using mu) for actual default likelihood. Practice structural credit models in our FRM Part II Credit Risk question bank.

How does the Merton model work for measuring credit risk, and what does the structural diagram look like?

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