FRM Exam Part II · Credit Value at Risk

Merton Model and Distance to Default Explained

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. The larger it is, the lower the default probability. In the plain Merton model, PD = N(−DD).

Q: How is KMV different from the Merton model?

KMV uses the same asset-versus-debt logic but sets the default point from short-term and long-term liabilities. It then maps distance to default to an expected default frequency using historical default data. This avoids relying on the normal distribution.

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

Structural models explain default through the firm's asset value falling below its debt. Reduced-form models treat default as a surprise event with a hazard rate, calibrated to market prices such as credit spreads. Structural models need balance sheet data. Reduced-form models do not.

Q: How do you get asset value and volatility in practice?

You observe equity value and equity volatility. You then solve two equations together: the equity call formula and σE·E = N(d1)·σV·V. The solution gives the unobservable asset value and asset volatility.

Updated 11 October 2026 · Fact-checked

The Merton model treats a firm's equity as a call option on its assets, with debt face value as the strike. Default happens if assets fall below debt at maturity. Distance to default measures how many standard deviations assets sit above debt. Default probability is N(−d2). KMV maps distance to default to an empirical EDF.

Understand Structural Models: Merton and KMV

A structural model explains default using the firm's balance sheet. The firm has assets worth V. It owes debt with face value D, due at time T. Asset value moves randomly. If V is below D at T, the firm defaults.

Merton's insight is that shareholders hold a call option on the firm's assets with strike D. If assets exceed D at T, shareholders pay the debt and keep the rest. If assets are below D, they walk away (limited liability) and get zero. So equity E = Black-Scholes call value. Debt holders are then long a risk-free bond and short a put option on assets.

Assets are assumed to follow a lognormal process. The probability that V_T < D under the risk-neutral measure is N(−d2). Using the real-world asset drift μ instead of the risk-free rate r gives the real-world default probability. Distance to default (DD) is the number of standard deviations between expected asset value and the default point. A larger DD means lower default probability.

Asset value and asset volatility cannot be observed. You only see equity value and equity volatility. Merton links them: E = V·N(d1) − D·e^(−rT)·N(d2), and σE·E = N(d1)·σV·V. Solving these two equations together gives V and σV.

KMV (Moody's KMV) makes the model practical. It uses a default point, often short-term debt plus half of long-term debt, rather than total debt. It computes DD, then maps DD to an expected default frequency (EDF) using a large historical database of defaults, instead of relying on the normal distribution. Compare this with reduced-form models, which do not model assets. They treat default as a random event arriving at a hazard rate, calibrated to market prices such as credit spreads.

Key formulas to remember

Equity as a call option: E = V·N(d1) − D·e^(−rT)·N(d2); V = asset value, D = face value of zero-coupon debt, r = risk-free rate, T = maturity.
d1 and d2: d1 = [ln(V ÷ D) + (r + σV²÷2)·T] ÷ (σV·√T); d2 = d1 − σV·√T; Uses asset volatility σV, not equity volatility.
Risk-neutral default probability: PD = N(−d2); Probability that V_T < D under the risk-neutral measure.
Link between equity and asset volatility: σE·E = N(d1)·σV·V; Used with the call formula to solve for V and σV.
Value of risky debt: B = V − E = D·e^(−rT) − Put(V, D); Debt equals risk-free bond minus a put on assets.
Distance to default (real-world): DD = [ln(V ÷ D) + (μ − σV²÷2)·T] ÷ (σV·√T); μ = expected asset return. Real-world PD = N(−DD). KMV maps DD to an empirical EDF instead.
Credit spread in Merton: s = −(1 ÷ T)·ln(B ÷ D) − r; B is the market value of risky debt, so the spread is the yield on B minus r.

How to solve Structural Models: Merton and KMV questions

Most questions give you assets, debt, volatility and rate, and ask for DD, PD, or a value. Follow these steps.

1Identify what is given: asset value V (or equity E), debt D, maturity T, σV (or σE), and whether r or μ is supplied.
2Check whether the question wants a risk-neutral or real-world probability. Risk-neutral uses r, real-world uses μ.
3Compute the numerator: ln(V ÷ D) plus the drift term with the correct sign on σV²÷2.
4Divide by σV·√T to get d2 or DD. Get d1 by adding σV·√T if needed.
5Convert to probability with N(−d2) or N(−DD). Use the symmetry N(−x) = 1 − N(x).
6If only equity data is given, say that V and σV must be solved from the two Merton equations. Do not use σE as σV.
7For debt or spread questions, use B = V − E and compare B with D·e^(−rT).
8State the interpretation: a higher DD means a lower default probability.

Quickest way: Fast DD shortcut

When to use it: Use when the question gives V, D, σV and a short horizon, and the options are spread apart.

Compute ln(V ÷ D). For small gaps, ln(V ÷ D) ≈ (V − D) ÷ D.
Add the drift term (r or μ minus σV²÷2) times T.
Divide by σV·√T. This is DD.
Recall N(−1) ≈ 0.159, N(−1.645) ≈ 0.05, N(−2) ≈ 0.023, N(−3) ≈ 0.0013.
Pick the nearest option and check that higher DD gives lower PD.

Common mistakes in Structural Models: Merton and KMV

Using equity volatility in d1 and d2.
Equity volatility is the only one you can observe, so it feels natural to use.
Fix: The formulas use asset volatility σV. Asset volatility is lower than equity volatility because of leverage.
Treating N(d2) as the default probability.
N(d2) is the call-style exercise probability, which looks like the headline output.
Fix: N(d2) is the probability of no default. Default probability is N(−d2) = 1 − N(d2).
Mixing risk-neutral and real-world probabilities.
Both use the same formula shape with a different drift.
Fix: Use r for risk-neutral PD and valuation. Use μ for real-world PD or DD.
Assuming KMV EDF equals N(−DD).
Students stop after computing DD in the Merton framework.
Fix: KMV maps DD to EDF through an empirical default database. The normal assumption is dropped at that step.
Confusing structural and reduced-form models.
Both produce default probabilities and spreads.
Fix: Structural models tie default to asset value versus debt. Reduced-form models use an exogenous hazard rate and need no asset data.
Forgetting that Merton debt is a single zero-coupon bond.
Real firms have many debt issues and coupons.
Fix: Remember the assumption: one zero-coupon liability, default only at maturity. Merton also gives low short-term spreads, which is a known weakness.

Worked examples

Example 1

A firm has asset value ₹1,200 crore and zero-coupon debt of ₹1,000 crore due in 1 year. Asset volatility is 20%, the risk-free rate is 5%. Find d2 and the risk-neutral default probability. Use N(0.7) = 0.758, N(0.8) = 0.788 as needed.

Show the solution

ln(1200 ÷ 1000) = ln(1.2) = 0.1823.
Drift term: (0.05 − 0.20²÷2) × 1 = 0.05 − 0.02 = 0.03.
Numerator = 0.1823 + 0.03 = 0.2123.
Denominator = 0.20 × 1 = 0.20.
d2 = 0.2123 ÷ 0.20 = 1.0615.
Using N(1.06) ≈ 0.855, PD = N(−1.0615) ≈ 1 − 0.855 = 0.145.

Answer: d2 ≈ 1.06, risk-neutral default probability ≈ 14.4% to 14.5%, about 14.5%.

Example 2

A firm has asset value $500 million, debt due in 1 year of $400 million, risk-free rate 4%. Equity value is $140 million. Which is the Merton market value of the risky debt, and is it above or below the risk-free value of debt, using e^(−0.04) = 0.9608?

Show the solution

Debt value B = V − E = 500 − 140 = $360 million.
Risk-free value of the debt = 400 × 0.9608 = $384.3 million.
Compare: 360 is below 384.3.
The gap, about $24.3 million, is the value of the put option on assets that debt holders have sold.

Answer: Risky debt is worth $360 million, about $24.3 million below the risk-free value of $384.3 million.

Exam tips

Know the direction of every relationship: higher σV, higher leverage or longer T with weak assets raise PD for a given asset value, and a higher DD lowers PD.
Expect conceptual MCQs on the option analogy: equity is a call, risky debt is a risk-free bond minus a put.
When asked about KMV, state three points: default point, DD, and the empirical mapping to EDF.
Know the limitations: single zero-coupon debt, default only at maturity, unobservable asset values, lognormal assets, and understated short-term spreads.
Read carefully whether the question asks for risk-neutral or real-world probability before choosing r or μ.

Practice questions from Credit Value at 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 by which expected asset value exceeds the default point at the horizon. The larger it is, the lower the default probability. In the plain Merton model, PD = N(−DD).

How is KMV different from the Merton model?

KMV uses the same asset-versus-debt logic but sets the default point from short-term and long-term liabilities. It then maps distance to default to an expected default frequency using historical default data. This avoids relying on the normal distribution.

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

Structural models explain default through the firm's asset value falling below its debt. Reduced-form models treat default as a surprise event with a hazard rate, calibrated to market prices such as credit spreads. Structural models need balance sheet data. Reduced-form models do not.

How do you get asset value and volatility in practice?

You observe equity value and equity volatility. You then solve two equations together: the equity call formula and σE·E = N(d1)·σV·V. The solution gives the unobservable asset value and asset volatility.