How does the Merton model calculate distance to default and what are its limitations?

The Merton model (1974) is the foundation of structural credit risk analysis. It uses option pricing theory to model default as the event where a firm's asset value falls below its debt obligations.

Core Idea:

• A firm has assets (V) and debt (D) maturing at time T
• Equity = Call option on assets with strike = D
• If V_T < D at maturity, the firm defaults (assets can't cover debt)
• Equity holders get max(V_T - D, 0)
• Debt holders get min(V_T, D)

Distance to Default (DD):

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

Where:

• V = current asset value
• D = default point (face value of debt)
• mu = expected asset return
• sigma_V = asset volatility
• T = time horizon

DD measures how many standard deviations the firm's assets are above the default point. Higher DD = lower default probability.

Worked Example — Redstone Manufacturing:

Redstone has:

• Market value of assets: $850M
• Short-term debt + 0.5 x long-term debt (default point): $600M
• Expected asset return: 8%
• Asset volatility: 25%
• Time horizon: 1 year

DD = [ln(850/600) + (0.08 - 0.5 x 0.0625) x 1] / (0.25 x 1)

= [0.3483 + 0.04875] / 0.25

= 0.3971 / 0.25

= 1.588

Using the normal distribution: P(default) = N(-1.588) = 5.6%

The Problem: We Can't Observe V or sigma_V

Asset value and asset volatility aren't directly observable. The Merton model solves this using two equations:

1. E = V x N(d1) - D x e^{-rT} x N(d2) (Black-Scholes for equity value)
2. sigma_E x E = N(d1) x sigma_V x V (relates equity vol to asset vol)

You observe E (market cap) and sigma_E (equity volatility), then solve simultaneously for V and sigma_V.

Limitations:

1. Assumes a single debt maturity — real firms have complex capital structures with bonds, loans, revolvers, and leases maturing at different times
2. Asset returns assumed lognormal — ignores jumps and fat tails
3. Default only at maturity T — in reality, firms can default at any time (first-passage models address this)
4. Static capital structure — firms issue new debt, buy back shares, and restructure
5. Credit spreads too low — the model systematically underestimates credit spreads for investment-grade firms

Despite these limitations, the Merton framework remains foundational because it establishes the link between equity markets and credit risk. The KMV model extends it significantly.

Explore structural credit models in depth in our FRM Part II course.