FRM Exam Part II · Estimating Default Probabilities

Merton Model and Distance to Default Explained

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 counts how many standard deviations assets sit above debt. Default probability is N(−d2) under risk-neutral drift, or N(−DD) with real-world drift.

Understand Merton Model and Equity-Based Default Estimation

The Merton model is a structural model. It explains default using the firm's balance sheet. The firm has assets worth V. It has issued one zero-coupon bond with face value D due at time T. Nothing else is assumed to happen before T.

At T, there are two outcomes. If V is greater than D, the firm repays debt and shareholders keep V − D. If V is less than D, the firm defaults and shareholders get nothing, because of limited liability. So equity payoff = max(V − D, 0). That is the payoff of a call option on firm assets with strike D.

This gives the pricing link. Equity E = V·N(d1) − D·e^(−rT)·N(d2). Risky debt is a risk-free bond minus a put option on assets. Under risk-neutral valuation, the probability of default is N(−d2). With a real-world asset drift μ instead of r, you replace r by μ in d2 to get the real-world default probability.

The practical problem is that asset value V and asset volatility σV cannot be observed. Equity value E and equity volatility σE can. So you solve two equations: the option formula for E, and the link σE·E = N(d1)·σV·V. This gives V and σV. KMV-style methods do this, define a default point (short-term debt plus part of long-term debt), and compute distance to default as DD = (V − default point) ÷ (V·σV), or the log version. KMV then maps DD to an expected default frequency (EDF) using an empirical database, not the normal curve.

Compare with reduced-form models. These do not model assets. They treat default as a surprise event driven by a hazard rate calibrated to market prices such as credit spreads. Structural models give economic intuition. Reduced-form models fit market prices more easily.

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 debt, r = risk-free rate, T = maturity, N = standard normal CDF.
d1 and d2: d1 = [ln(V ÷ D) + (r + σV²÷2)·T] ÷ (σV·√T); d2 = d1 − σV·√T; σV is asset volatility. Use the same inputs in both.
Risk-neutral default probability: PD = N(−d2); This is the probability that V is below D at T under the risk-neutral measure.
Real-world default probability: PD = N(−d2*), with d2* = [ln(V ÷ D) + (μ − σV²÷2)·T] ÷ (σV·√T); μ is the expected asset return. Use it for real-world PD.
Equity volatility link: σE·E = N(d1)·σV·V; Used with the equity formula to solve for the unobservable V and σV.
Distance to default (simple): DD = (V − default point) ÷ (V·σV); Number of asset standard deviations above the default point, over one year. Default point is often short-term debt + 50% of long-term debt in KMV.
Risky debt value: B = V − E = D·e^(−rT) − Put(V, D); Debt holders are long a risk-free bond and short a put on assets.

How to solve Merton Model and Equity-Based Default Estimation questions

Use this order for any Merton or distance-to-default question.

1Identify what is given: asset value V, debt D (or default point), asset volatility σV, drift, horizon T. If only equity data is given, you must first back out V and σV.
2Decide the measure. Risk-neutral PD uses r. Real-world PD uses μ. The question usually states which.
3Compute the ratio V ÷ D (or V ÷ default point) and its natural log.
4Compute d2 (or DD) with the right drift term: (drift − σV²÷2)·T in the numerator, σV·√T in the denominator.
5Convert to probability: PD = N(−d2). Read N(·) from standard normal values, such as N(−1) ≈ 0.1587 and N(−2) ≈ 0.0228.
6If the question uses KMV, note that the EDF comes from an empirical map of DD, not the normal CDF.
7State the interpretation: higher leverage, higher volatility or lower asset value lowers DD and raises PD.

Quickest way: Shortcut with simple distance to default

When to use it: Use when the question gives DD as a number of standard deviations, or gives V, default point and σV, and asks for a normal-based PD.

Compute DD = (V − default point) ÷ (V·σV).
PD ≈ N(−DD).
Remember: N(−1) ≈ 15.9%, N(−1.645) ≈ 5%, N(−2) ≈ 2.3%, N(−3) ≈ 0.13%.
Eliminate options that move the wrong way: more leverage or more volatility must raise PD.

Common mistakes in Merton Model and Equity-Based Default Estimation

Using N(d2) as the default probability.
N(d2) appears in the equity formula, so it feels like the key number.
Fix: N(d2) is the probability that assets finish above D. Default probability is N(−d2) = 1 − N(d2).
Calling equity a put option on assets.
Debt is linked to a put, which confuses the two.
Fix: Equity is a call on assets with strike D. Debt holders are short a put on assets.
Using r for a real-world PD, or μ for a risk-neutral PD.
Both versions use the same formula shape.
Fix: Risk-neutral uses r and is used for pricing. Real-world uses μ and is used for risk and capital. Real-world PD is usually lower when μ is greater than r.
Forgetting the −σV²÷2 term or the √T in the denominator.
Memorising the formula without understanding where it comes from.
Fix: Assets are lognormal, so the expected log return is drift minus σ²÷2. Standard deviation scales with √T.
Treating KMV EDF as N(−DD).
Textbook Merton and KMV are taught together.
Fix: KMV maps DD to EDF with historical default data. The normal CDF is only the pure Merton result.
Plugging equity volatility into d2 as if it were asset volatility.
Equity volatility is observable, asset volatility is not.
Fix: Equity volatility is higher than asset volatility for a levered firm. Use σE·E = N(d1)·σV·V to solve for σV.

Worked examples

Example 1

A firm has asset value V = $120 million and a zero-coupon debt of face value D = $100 million due in 1 year. Asset volatility is 20%. The expected asset return is 8% (real-world). Estimate the real-world default probability. Use ln(1.2) = 0.1823, N(−1.0) = 0.1587, N(−1.1) = 0.1357, N(−0.9) = 0.1841.

Show the solution

ln(V ÷ D) = ln(1.2) = 0.1823.
Drift term: (μ − σV²÷2)·T = (0.08 − 0.02)·1 = 0.06.
Numerator = 0.1823 + 0.06 = 0.2423.
Denominator = σV·√T = 0.20.
d2* = 0.2423 ÷ 0.20 = 1.2115 ≈ 1.21.
PD = N(−1.21) ≈ 0.113, using N(−1.2) ≈ 0.1151 and N(−1.25) ≈ 0.1056 as a check.

Answer: About 11.3%, roughly 11%.

Example 2

A bank uses a KMV-style approach. A firm has asset value ₹5,000 crore, asset volatility 25% a year, short-term debt ₹1,500 crore and long-term debt ₹2,000 crore. The default point is short-term debt plus 50% of long-term debt. Find the distance to default and the normal-based PD. N(−2) = 0.0228, N(−1.5) = 0.0668, N(−2.5) = 0.0062.

Show the solution

Default point = 1,500 + 0.5 × 2,000 = ₹2,500 crore.
Asset gap = 5,000 − 2,500 = ₹2,500 crore.
Asset standard deviation = 0.25 × 5,000 = ₹1,250 crore.
DD = 2,500 ÷ 1,250 = 2.0.
Normal-based PD = N(−2.0) = 0.0228.

Answer: DD = 2.0 standard deviations; normal-based PD ≈ 2.3%. A KMV EDF would come from an empirical DD-to-EDF map and could differ.

Exam tips

Check the first word: risk-neutral or real-world. It decides which drift you use.
Expect interpretation questions: which change raises PD? Higher debt, higher σV and lower V all raise PD.
Know the call and put roles. Equity is a call on assets. Risky debt is risk-free debt minus a put.
Be ready to contrast structural and reduced-form models: asset-based and economic versus hazard-rate and market-calibrated.
Learn the limits: one debt maturity, default only at T, and unobservable asset values.

Practice questions from Estimating Default Probabilities

Merton Model and Equity-Based Default Estimation in other exams

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

Merton Model and Equity-Based Default Estimation: frequently asked questions

What is distance to default?

It measures how far firm assets are above the default point, in standard deviations of asset value. A larger distance means lower default risk. Pure Merton converts it to PD with the normal CDF, while KMV uses an empirical mapping to EDF.

How is the Merton model different from a reduced-form model?

The Merton model is structural. It links default to the firm's asset value falling below its debt. Reduced-form models treat default as an unexpected event with a hazard rate, usually calibrated to bond or CDS spreads. They need no asset value model.

Why is equity a call option on firm assets?

Shareholders have limited liability. If assets exceed debt at maturity they receive the excess. If not, they receive zero. That payoff, max(V − D, 0), matches a call with strike D.

Which default probability does N(−d2) give?

With r as the drift it gives the risk-neutral probability, used for pricing credit instruments. With the real asset return μ it gives the real-world probability, which is the one used for risk management.