FRM Exam Part II · Introduction to Credit Risk Modeling and Assessment

Merton Model and KMV Structural Credit Models

Q: What is distance to default in the Merton model?

It is the number of standard deviations by which expected asset value exceeds the default point at the horizon. A higher DD means lower default risk. In the pure model PD = N(−DD).

Q: How is KMV different from the Merton model?

KMV uses the Merton framework but sets the default point from short- and long-term debt and estimates asset value and volatility from equity data. It then maps DD to an EDF using historical default data instead of the normal distribution.

Q: What is the difference between structural and reduced-form credit models?

Structural models say default occurs when asset value falls below debt, so default is linked to firm fundamentals. Reduced-form models treat default as a random event with a hazard rate calibrated to market prices, such as bond or CDS spreads.

Q: Why is equity a call option on firm assets?

Shareholders have limited liability. At maturity they receive assets minus debt if positive, and zero otherwise. That payoff is max(V − D, 0), a call with strike D.

Updated 11 October 2026 · Fact-checked

The Merton model treats a firm's equity as a European call option on its assets, with strike equal to the face value of zero-coupon debt. Default happens if assets are below debt at maturity. Risk-neutral default probability is N(−d2). KMV maps distance to default to an empirical default frequency (EDF).

Understand Structural Models: Merton Model and KMV

A structural model explains default using the firm's balance sheet. Default is not a random surprise. It happens when asset value falls below what the firm owes.

Merton (1974) makes this exact. Assume the firm has asset value V, which follows a lognormal process with volatility σ_V. It has one zero-coupon debt issue with face value D due at time T. At T, shareholders get max(V − D, 0). That is a call option on assets with strike D. Debt holders get D − max(D − V, 0), which is risk-free debt minus a put option sold to shareholders.

Because equity is a call, Black-Scholes applies. The probability the call finishes in the money is N(d2), so the risk-neutral default probability is N(−d2). Here d2 = [ln(V/D) + (r − σ²/2)T] ÷ (σ√T). Using the real-world asset drift μ instead of r gives a real-world default probability.

Distance to default (DD) counts how many standard deviations the asset value sits above the default point. A larger DD means lower default risk. In the pure model with a normal assumption, PD = N(−DD).

KMV (Moody's KMV) makes this practical. Asset value and asset volatility are not observed, so they are backed out from equity value and equity volatility. The default point is usually short-term debt plus a fraction of long-term debt. DD is then mapped to an expected default frequency (EDF) using a large database of historical defaults, because the normal assumption understates real default rates. In contrast, reduced-form models do not model assets. They treat default as an unpredictable event driven by a hazard rate fitted to market prices.

Key formulas to remember

Equity as a call: E = V·N(d1) − D·e^(−rT)·N(d2); Black-Scholes applied to assets V with strike D.
d1 and d2: d1 = [ln(V/D) + (r + σ_V²/2)T] ÷ (σ_V√T); d2 = d1 − σ_V√T; Use r for risk-neutral, μ in place of r for real-world.
Default probability: PD = N(−d2); Probability that V < D at time T in the Merton model.
Distance to default (simple): DD = [ln(V/D) + (μ − σ_V²/2)T] ÷ (σ_V√T); Number of standard deviations from the default point.
Risky debt value: B = V − E = D·e^(−rT) − Put(V, D); Debt equals risk-free debt minus a put on assets.
Credit spread: s = −(1/T)·ln(B ÷ (D·e^(−rT))); Yield on risky debt minus r, with B the value of the debt.
Equity volatility link: σ_E·E = N(d1)·σ_V·V; Used to solve for unobserved V and σ_V.

How to solve Structural Models: Merton Model and KMV questions

Use this order for most Merton and KMV questions.

1Identify the inputs: asset value V, debt face value D (default point), maturity T, volatility, and the rate (r or μ).
2Check what is asked: risk-neutral PD, real-world PD, DD, debt value or spread.
3If V and σ_V are given, compute d2 (or DD). If only equity data are given, use the two equations E and σ_E to infer V and σ_V.
4Convert to probability with the normal table: PD = N(−d2) or N(−DD).
5For debt value or spread, compute V − E, or D·e^(−rT) minus the put, then the spread from the formula.
6For KMV, remember DD is mapped to an EDF empirically, not through the normal table.
7Sanity check: higher leverage, higher volatility or longer horizon with weak assets should raise PD and spread.

Quickest way: Shortcut: compute DD, then N(−DD)

When to use it: When the question gives V, D, σ_V, T and a drift, and asks for PD or DD.

Compute ln(V/D).
Add (μ − σ²/2)·T and divide by σ√T to get DD.
Read N(−DD) from common values: N(−1)=15.9%, N(−2)=2.3%, N(−3)=0.13%.
Eliminate options that move the wrong way when leverage or volatility changes.

Common mistakes in Structural Models: Merton Model and KMV

Using N(d2) as the default probability.
N(d2) is the call's in-the-money probability, which is the chance of survival.
Fix: Default probability is N(−d2) = 1 − N(d2).
Mixing risk-neutral and real-world drift.
Both versions use the same formula with different drift.
Fix: Use r for risk-neutral PD and pricing spreads. Use μ for real-world PD and DD.
Saying KMV EDF equals N(−DD).
The textbook Merton model gives that link.
Fix: KMV maps DD to EDF using historical default experience, which captures fat tails and real default rates.
Treating equity volatility as asset volatility.
Equity volatility is the observable number.
Fix: Asset volatility is lower: σ_V = σ_E·E ÷ (N(d1)·V) in the model.
Confusing structural and reduced-form models.
Both produce default probabilities and spreads.
Fix: Structural models link default to asset value versus debt. Reduced-form models use an exogenous hazard rate and no balance sheet.
Forgetting the model assumes one zero-coupon debt issue and default only at maturity.
Real firms have coupons and many maturities.
Fix: State this as a limitation, along with constant volatility and unobservable assets.

Worked examples

Example 1

A firm has asset value V = $120 million, zero-coupon debt face D = $100 million due in 1 year, asset volatility 20% and expected asset return μ = 8.9%. Compute the distance to default and the real-world default probability. (ln 1.2 = 0.1823; N(−1.5) = 0.0668; N(−1.0)=0.1587; N(−2.0)=0.0228)

Show the solution

ln(V/D) = ln(1.2) = 0.1823.
Drift term: (μ − σ²/2)T = 0.089 − 0.02 = 0.069.
Numerator = 0.1823 + 0.069 = 0.2513.
Denominator = 0.20 × 1 = 0.20.
DD = 0.2513 ÷ 0.20 = 1.2565, about 1.26.
PD = N(−1.26) ≈ 0.104, about 10.4%.

Answer: DD ≈ 1.26 standard deviations; real-world PD ≈ 10.4%.

Example 2

A firm has V = $150 million, zero-coupon debt D = $100 million due in 1 year, risk-free rate 5% (continuous) and the model gives equity E = $60 million. Find the market value of the debt and the credit spread. (e^−0.05 = 0.9512)

Show the solution

Debt value B = V − E = 150 − 60 = $90 million.
Risk-free value of debt = 100 × 0.9512 = $95.12 million.
Ratio B ÷ risk-free value = 90 ÷ 95.12 = 0.9462.
ln(0.9462) = −0.0553.
Spread = −(1/1) × (−0.0553) = 5.53%, about 5.5%.

Answer: Debt is worth $90 million and the credit spread is about 5.5% (about 553 bps).

Exam tips

Know that N(−d2) is risk-neutral default probability and N(d2) is survival.
Expect qualitative questions on how leverage, asset volatility and maturity change PD, DD and spread.
Be ready to explain why KMV uses empirical EDF and why asset volatility is lower than equity volatility.
Learn the structural versus reduced-form contrast: asset-based and endogenous default versus hazard-rate and exogenous default.
State Merton limitations quickly: single zero-coupon debt, default only at maturity, lognormal assets, constant volatility.

Practice questions from Introduction to Credit Risk Modeling and Assessment

Structural Models: Merton Model and KMV in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Structural Models: Merton Model and KMV: frequently asked questions

What is distance to default in the Merton model?

It is the number of standard deviations by which expected asset value exceeds the default point at the horizon. A higher DD means lower default risk. In the pure model PD = N(−DD).

How is KMV different from the Merton model?

KMV uses the Merton framework but sets the default point from short- and long-term debt and estimates asset value and volatility from equity data. It then maps DD to an EDF using historical default data instead of the normal distribution.

What is the difference between structural and reduced-form credit models?

Structural models say default occurs when asset value falls below debt, so default is linked to firm fundamentals. Reduced-form models treat default as a random event with a hazard rate calibrated to market prices, such as bond or CDS spreads.

Why is equity a call option on firm assets?

Shareholders have limited liability. At maturity they receive assets minus debt if positive, and zero otherwise. That payoff is max(V − D, 0), a call with strike D.