FRM Exam Part II · Portfolio Credit Risk

Merton Model, Distance to Default and KMV EDF

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

It is the number of standard deviations between the firm's expected asset value at the horizon and its default point. A higher distance means a lower chance of default. Under normal returns, PD equals N(−DD).

Q: How does KMV EDF differ from the Merton default probability?

Merton uses the normal distribution to turn distance to default into a probability. KMV uses a database of past defaults to map distance to default into an EDF. This captures fat tails that the normal curve misses.

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

Structural models, such as Merton, link default to the firm's asset value falling below its debt. Reduced-form models treat default as a random event with a hazard rate, calibrated to market prices. Structural models give economic insight, while reduced-form models fit spreads more easily.

Q: Why do we solve for asset value and volatility in the Merton model?

Asset value and asset volatility cannot be observed directly. You can observe equity value and equity volatility. Two equations link them, and you solve both together for V and σ_V.

Updated 11 October 2026 · Fact-checked

The Merton model treats a firm's equity as a call option on its assets, with debt as the strike. Default happens if asset value falls below debt at maturity. Distance to default measures how many standard deviations assets sit above the default point. KMV maps that distance to an empirical default frequency (EDF).

Understand Structural Models: Merton and KMV

A structural model explains default using the firm's balance sheet. The firm has assets of value V. It has issued a single zero-coupon bond with face value D due at time T. Asset value moves randomly, usually as a geometric Brownian motion with volatility σ_V.

At time T, if V is above D, shareholders pay the debt and keep the rest. If V is below D, shareholders walk away and lenders take the assets. So equity is worth max(V − D, 0). That is the payoff of a European call option on the assets with strike D. Debt equals assets minus equity, so lenders hold a risk-free bond and are short a put option on the assets.

Under risk-neutral pricing, the Black-Scholes-Merton formula gives equity value. The risk-neutral probability of default is N(−d2), where d2 = [ln(V ÷ D) + (r − σ_V²÷2)T] ÷ (σ_V√T). For the real-world default probability, you use the asset's expected return μ in place of r in the d2 expression. The result is N(−DD). Option and equity values are still computed with r.

Distance to default (DD) is the number of standard deviations between expected asset value and the default point. A larger DD means lower default risk. The simple form is DD = [ln(V ÷ D) + (μ − σ_V²÷2)T] ÷ (σ_V√T). Under normality, PD = N(−DD).

The problem: V and σ_V cannot be observed. You see equity value E and equity volatility σ_E. The link is σ_E × E = N(d1) × σ_V × V, plus the option pricing equation. You solve both together for V and σ_V. KMV goes further. It uses a default point (short-term debt plus half of long-term debt) and then maps DD to an expected default frequency (EDF) using a large database of historical defaults. This avoids relying on the normal distribution, which understates default frequency at large DD. Reduced-form models differ: they treat default as an unpredictable event with a hazard rate, not a result of asset value.

Key formulas to remember

Equity as a call option: E = V × N(d1) − D × e^(−rT) × N(d2); Equity is a call on assets V with strike D (face value of zero-coupon debt).
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 values. Use μ in place of r for real-world probability.
Risk-neutral default probability: PD = N(−d2); Probability that V < D at time T under the risk-neutral measure.
Distance to default: DD = [ln(V ÷ D) + (μ − σ_V²÷2)T] ÷ (σ_V√T); Number of standard deviations from expected log asset value to the default point.
Real-world default probability: PD = N(−DD); Valid only if asset returns are normal in logs. KMV replaces this with an empirical EDF.
Equity and asset volatility link: σ_E × E = N(d1) × σ_V × V; Used with the equity formula to solve for unobserved V and σ_V.
Value of risky debt: B = V − E = D × e^(−rT) − Put(V, D); Lenders are long a risk-free bond and short a put on the assets.
KMV default point: Default point = short-term debt + 0.5 × long-term debt; Used instead of total debt in KMV's DD.
Simplified KMV distance to default: DD = (V − default point) ÷ (V × σ_V); Approximation for a one-year horizon. It ignores drift and replaces ln(V ÷ default point) with (V − default point) ÷ V, so it is the gap as a share of assets, over asset volatility.

How to solve Structural Models: Merton and KMV questions

Use this order for any Merton or KMV question. It keeps you from mixing risk-neutral and real-world numbers.

1Identify what is given: asset value V or equity E, debt D, volatility, horizon T, and the rate r or expected return μ.
2Decide the measure. If the question asks for a price or risk-neutral PD, use r. If it asks for real-world PD or DD, use μ.
3If V and σ_V are not given, remember they come from equity data. Check if the question supplies them directly or expects you to use σ_E × E = N(d1) × σ_V × V.
4Compute ln(V ÷ D), then the drift term, then divide by σ_V√T to get DD or d2.
5Convert to probability with N(−DD) or N(−d2). Use the normal table values the question provides.
6If KMV is mentioned, use the default point (short-term debt plus half of long-term debt) and note that EDF comes from empirical mapping, not N(·).
7State the interpretation: higher DD means lower default risk, and equity is a call, debt is risk-free debt minus a put.

Quickest way: Fast DD check for multiple choice

When to use it: Use when options give DD or PD values and you have under two minutes.

Approximate ln(V ÷ D) with the natural log. For ratios near 1, ln(x) ≈ x − 1.
Compute the drift term (μ − σ²÷2)T. Often small, so check whether it changes the answer.
Divide by σ_V√T. This is your DD.
Match to the option. Reasonable PD values: DD of 1 gives about 15.9%, DD of 2 about 2.3%, DD of 3 about 0.13%.
Eliminate options that break logic: PD cannot rise as DD rises.
Use this as a conditional check when a question compares the two measures with the same V, D, σ_V and T. Compare N(−d2) with N(−DD). If μ is above r, DD is larger than d2, so N(−d2) is larger than N(−DD): the risk-neutral PD is higher than the real-world PD. If μ is below r, the order reverses.

Common mistakes in Structural Models: Merton and KMV

Using r when the question asks for real-world default probability.
Black-Scholes habit makes r the default input.
Fix: Real-world PD and DD use the expected asset return μ. N(−d2) with r is the risk-neutral PD.
Forgetting the −σ²÷2 term in the drift.
Students remember ln(V ÷ D) and the rate but drop the volatility correction.
Fix: Write the full numerator every time: ln(V ÷ D) + (μ − σ_V²÷2)T.
Plugging equity volatility in place of asset volatility.
Equity volatility is the number you can observe, so it looks like the right input.
Fix: Equity is leveraged, so σ_E is higher than σ_V. Use σ_E × E = N(d1) × σ_V × V to get σ_V.
Saying lenders hold a call option.
Confusion over who owns which payoff.
Fix: Shareholders are long a call on assets. Lenders are long risk-free debt and short a put on assets.
Treating KMV EDF as N(−DD).
The textbook Merton model gives that result, so students carry it over.
Fix: KMV maps DD to EDF using historical default data. That corrects for fat tails and non-normal asset returns.
Using total debt as the KMV default point.
Merton assumes one zero-coupon bond.
Fix: KMV uses short-term debt plus half of long-term debt, because long debt is rarely due within the horizon.

Worked examples

Example 1

A firm has asset value V = 120, zero-coupon debt face value D = 100 due in 1 year, asset volatility σ_V = 25%, and expected asset return μ = 9%. Compute the distance to default and the real-world PD using N(−0.96) ≈ 0.168 as given.

Show the solution

ln(120 ÷ 100) = ln(1.2) = 0.1823.
Drift term: (0.09 − 0.25²÷2) × 1 = 0.09 − 0.03125 = 0.05875.
Numerator = 0.1823 + 0.05875 = 0.2411.
Denominator = 0.25 × √1 = 0.25.
DD = 0.2411 ÷ 0.25 = 0.964, which rounds to 0.96.
PD = N(−0.96) ≈ 0.168, or 16.8%.

Answer: DD ≈ 0.96 standard deviations; real-world PD ≈ 16.8%. The firm is only about one standard deviation above default, so risk is high.

Example 2

Under the Merton model, a firm's debt is a 1-year zero-coupon bond with face value 100 and r = 4%. Asset value is 150 and asset volatility is σ_V = 30%. Use N(1.63) = 0.9484 and N(1.33) = 0.9082. Find the equity value and the risk-neutral PD.

Show the solution

ln(150 ÷ 100) = ln(1.5) = 0.4055.
d1 = [0.4055 + (0.04 + 0.30²÷2) × 1] ÷ (0.30 × √1) = (0.4055 + 0.085) ÷ 0.30 = 1.6349, which is 1.63 to two decimals, the value in the table given.
d2 = d1 − σ_V√T = 1.6349 − 0.30 = 1.3349, which is 1.33 to two decimals.
The table values are for d1 = 1.63 and d2 = 1.33, not the unrounded 1.6349 and 1.3349. All results below are therefore approximate.
Risk-neutral PD = N(−d2) = 1 − 0.9082 = 0.0918, or about 9.18%.
Discount factor: e^(−0.04) = 0.9608.
Equity E = V × N(d1) − D × e^(−rT) × N(d2).
V × N(d1) = 150 × 0.9484 = 142.26.
D × e^(−rT) × N(d2) = 100 × 0.9608 × 0.9082 = 87.26.
E = 142.26 − 87.26 = 55.00.
Debt value = V − E = 150 − 55.00 = 95.00.

Answer: Using the rounded table values, equity ≈ 55.00, risk-neutral PD ≈ 9.18%, and risky debt is worth about 95.00, below the risk-free value of 96.08. These figures are approximate because d1 and d2 were rounded to 1.63 and 1.33.

Exam tips

Read the question for r versus μ. Many wrong options are built from using the wrong one.
Know the direction of every effect: higher V, lower σ_V or lower D raises DD and lowers PD. Higher volatility raises the value of equity as a call.
Expect a conceptual question on why KMV uses an empirical EDF. The answer is that real asset returns are not normal, so N(−DD) misstates default frequency.
Be ready to contrast structural and reduced-form models: structural links default to firm value; reduced-form uses an exogenous hazard rate and fits market spreads more easily.
Know the Merton limits: one debt maturity, default only at T, constant volatility, and unobservable asset value.

Practice questions from Portfolio Credit Risk

Structural Models: Merton 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 and KMV: frequently asked questions

What is distance to default in the Merton model?

It is the number of standard deviations between the firm's expected asset value at the horizon and its default point. A higher distance means a lower chance of default. Under normal returns, PD equals N(−DD).

How does KMV EDF differ from the Merton default probability?

Merton uses the normal distribution to turn distance to default into a probability. KMV uses a database of past defaults to map distance to default into an EDF. This captures fat tails that the normal curve misses.

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

Structural models, such as Merton, link default to the firm's asset value falling below its debt. Reduced-form models treat default as a random event with a hazard rate, calibrated to market prices. Structural models give economic insight, while reduced-form models fit spreads more easily.

Why do we solve for asset value and volatility in the Merton model?

Asset value and asset volatility cannot be observed directly. You can observe equity value and equity volatility. Two equations link them, and you solve both together for V and σ_V.