FRM Exam Part II · Credit Risk

Merton Model and Distance to Default for FRM Part II

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

It is the number of standard deviations by which the firm's asset value exceeds its default point over the horizon. The simple one-year formula uses the current asset value V. A larger distance means a lower probability of default. In KMV, DD is mapped to an expected default frequency using empirical data.

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

Structural models derive default from the firm's asset value falling below its debt. Reduced-form models treat default as a surprise event with a hazard rate, calibrated to bond or CDS prices. Structural models give economic insight; reduced-form models fit market data better.

Q: How does KMV differ from Merton?

KMV uses a default point based on short-term and part of long-term liabilities, and allows more realistic debt structures. It converts distance to default into an EDF using a historical default database rather than the normal distribution.

Q: Why is equity a call option in Merton?

Shareholders receive V − D if assets exceed debt at maturity and nothing otherwise, because of limited liability. That payoff is a call on assets with strike equal to the debt face value.

Updated 11 October 2026 · Fact-checked

Structural models, like Merton and KMV, treat default as the firm's asset value falling below its debt, so equity is a call option on assets. Distance to default measures how many standard deviations assets sit above the default point. Reduced-form models treat default as a random event with a hazard rate (intensity).

Understand Structural and Reduced-Form Default Models

A structural model explains default from the firm's balance sheet. The Merton model assumes the firm has one zero-coupon debt with face value D due at time T. At T, if asset value V is above D, the firm repays and shareholders keep V − D. If V is below D, shareholders walk away and lenders get V.

So equity is a European call option on the firm's assets with strike D. Risky debt equals a risk-free bond minus a put option on assets (the lenders have effectively sold a put to shareholders). Asset value is assumed lognormal, with drift μ and volatility σ.

In Merton, default means V_T < D. The real-world probability of default is N(−d2*), where d2* = [ln(V ÷ D) + (μ − σV²÷2)·T] ÷ (σV·√T). Here μ is the asset's expected return. The risk-neutral probability of default is N(−d2), where d2 is the Black-Scholes d2 with r replacing μ: d2 = [ln(V ÷ D) + (r − σV²÷2)·T] ÷ (σV·√T). Note that d2 has −σV²÷2 in the numerator, while d1 has +σV²÷2. Asset value and asset volatility are not observed. You back them out from equity value and equity volatility, using E = V·N(d1) − D·e^(−rT)·N(d2) and σE·E = N(d1)·σV·V.

KMV adapts Merton for practice. It defines a default point (short-term debt plus a share of long-term debt, often described as short-term liabilities plus half of long-term liabilities). Distance to default (DD) = (V − default point) ÷ (V·σV) for a one-year horizon, or the log-based version. KMV then maps DD to an expected default frequency (EDF) using a large empirical database of defaults, rather than the normal distribution.

Reduced-form models do not model the firm's assets. Default is an unpredictable event that arrives at a rate called the hazard rate or intensity λ. With a constant λ, the survival probability to time t is e^(−λt). The hazard rate is typically calibrated to bond or CDS spreads. Recovery is specified separately. These models fit market prices well but give no economic story for why default occurs.

Key formulas to remember

Equity as a call option: E = V·N(d1) − D·e^(−rT)·N(d2); V is asset value, D is face value of debt, r is the risk-free rate, T is maturity.
d1 and d2: d1 = [ln(V ÷ D) + (r + σV²÷2)·T] ÷ (σV·√T); d2 = d1 − σV·√T; Use σV, the asset volatility, not equity volatility.
Risk-neutral default probability: P(V_T < D) = N(−d2); For the real-world probability, replace r with μ in d2 (the numerator becomes ln(V ÷ D) + (μ − σV²÷2)·T).
Risky debt value: B = D·e^(−rT) − Put(V, D); Equals V − E. Put value is the credit risk cost.
Equity and asset volatility link: σE·E = N(d1)·σV·V; Used with the equity formula to solve for V and σV.
Distance to default (simple): DD = (V − default point) ÷ (V·σV), for a one-year horizon; Number of standard deviations of assets above the default point, using current asset value V over a one-year horizon.
Constant hazard rate survival: Q(t) = e^(−λt); PD(0,t) = 1 − e^(−λt); λ is the intensity of default.
Credit spread approximation: λ ≈ s ÷ (1 − R); s is the credit spread, R the recovery rate. An approximation.

How to solve Structural and Reduced-Form Default Models questions

Use this order for any structural or reduced-form question.

1Identify the model: asset-based (Merton, KMV) or intensity-based (hazard rate).
2For Merton, list V, D, σV, r and T. Check whether the volatility given is asset or equity volatility.
3If only equity data are given, remember you need to solve two equations for V and σV. Exam questions usually give V and σV directly or ask for the logic.
4Compute d1 and d2 with ln(V ÷ D), then N(−d2) for default probability.
5For KMV, find the default point, then DD = (V − default point) ÷ (V·σV) for a one-year horizon. Higher DD means lower default risk.
6For reduced-form, use Q(t) = e^(−λt) and PD = 1 − e^(−λt). Convert spreads with λ ≈ s ÷ (1 − R).
7State the interpretation: which measure (real-world or risk-neutral) and what assumption drives it.

Quickest way: Shortcut for distance to default and hazard rate questions

When to use it: When the question gives numbers and asks for DD, PD or a conceptual comparison.

For one-year DD with a simple formula, compute (V − default point) ÷ σ in rupee or dollar terms, where σ = V·σV. Do not use the normal table.
If asked for PD from DD, use N(−DD) as an approximation, noting that KMV uses empirical EDF instead.
For hazard rates, compute λ = s ÷ (1 − R) first, then PD = 1 − e^(−λt).
For concept questions: structural means asset value process and default threshold; reduced-form means exogenous intensity and market calibration.

Common mistakes in Structural and Reduced-Form Default Models

Using equity volatility in d1 and d2.
Equity volatility is observable, so it is tempting to plug in.
Fix: Use asset volatility σV. Equity volatility is higher for a levered firm: σE = N(d1)·σV·V ÷ E.
Treating N(d2) as the default probability.
d2 appears in the equity formula, so students mix up the tail.
Fix: N(d2) is the probability the firm repays (under the relevant measure). Default is N(−d2) = 1 − N(d2).
Mixing risk-neutral and real-world probabilities.
Both use the same formula with different drift.
Fix: Risk-neutral uses r. Real-world uses μ, the asset's expected return. Use real-world for risk management, risk-neutral for pricing.
Believing KMV EDF comes straight from the normal distribution.
The textbook Merton model does.
Fix: KMV maps DD to EDF using historical default frequencies, which captures fatter tails.
Saying reduced-form models explain why firms default.
Both families produce default probabilities.
Fix: Reduced-form default is a surprise event with intensity λ. It is calibrated to market prices, not derived from the balance sheet.
Forgetting that Merton assumes default only at maturity.
Real firms can default earlier.
Fix: Merton has one zero-coupon debt and default only at T. First-passage models allow earlier default.

Worked examples

Example 1

A firm has asset value of $100 million, one-year zero-coupon debt with face value $80 million, asset volatility of 20% and a risk-free rate of 5% (continuous). Using Merton, what is the risk-neutral probability of default? (Table values: N(−1.25) ≈ 0.1056, N(−1.30) ≈ 0.0968. Interpolate linearly.)

Show the solution

ln(100 ÷ 80) = ln(1.25) = 0.2231.
(r + σ²÷2)·T = 0.05 + 0.02 = 0.07.
d1 = (0.2231 + 0.07) ÷ 0.20 = 1.4657.
d2 = 1.4657 − 0.20 = 1.2657.
Interpolate between 1.25 and 1.30: the fraction is (1.2657 − 1.25) ÷ 0.05 = 0.314.
N(−1.2657) ≈ 0.1056 − 0.314 × (0.1056 − 0.0968) = 0.1056 − 0.314 × 0.0088 = 0.1056 − 0.0028 = 0.1028.

Answer: d2 ≈ 1.2657, so the risk-neutral default probability is N(−1.2657) ≈ 0.103, or about 10.3%.

Example 2

A bond has a credit spread of 3.0% a year and an assumed recovery rate of 40%. Using a constant hazard rate model, what is the approximate probability of default over 2 years? Use e^(−0.1) ≈ 0.9048.

Show the solution

λ ≈ s ÷ (1 − R) = 0.03 ÷ 0.60 = 0.05.
Survival over 2 years = e^(−0.05 × 2) = e^(−0.10) = 0.9048.
PD = 1 − 0.9048 = 0.0952.

Answer: About 9.5% (risk-neutral probability of default over 2 years).

Exam tips

Know the direction: higher asset volatility or higher leverage lowers DD and raises PD.
Expect conceptual questions on the differences: structural uses asset value and a default barrier; reduced-form uses an exogenous hazard rate calibrated to market spreads.
Read whether the question asks for risk-neutral or real-world probability, since the answer changes the drift used.
Remember the option view: equity is a call, risky debt is risk-free debt minus a put.
Credit spread ≈ λ × (1 − R) is an approximation; state it when you use it.

Practice questions from Credit Risk

Structural and Reduced-Form Default Models in other exams

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

Structural and Reduced-Form Default Models: frequently asked questions

What is distance to default in the Merton model?

It is the number of standard deviations by which the firm's asset value exceeds its default point over the horizon. The simple one-year formula uses the current asset value V. A larger distance means a lower probability of default. In KMV, DD is mapped to an expected default frequency using empirical data.

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

Structural models derive default from the firm's asset value falling below its debt. Reduced-form models treat default as a surprise event with a hazard rate, calibrated to bond or CDS prices. Structural models give economic insight; reduced-form models fit market data better.

How does KMV differ from Merton?

KMV uses a default point based on short-term and part of long-term liabilities, and allows more realistic debt structures. It converts distance to default into an EDF using a historical default database rather than the normal distribution.

Why is equity a call option in Merton?

Shareholders receive V − D if assets exceed debt at maturity and nothing otherwise, because of limited liability. That payoff is a call on assets with strike equal to the debt face value.