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FRM Exam Part I · Measuring Credit Risk

Merton Model Credit Risk: Structural Default Model Explained

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 fall below debt at maturity. You compute d2, then N(−d2) gives the risk-neutral default probability. Risky debt equals assets minus equity.

Understand Structural Models of Default (Merton Model)

A structural model explains default from the firm's balance sheet. Merton assumes the firm has assets worth V, which follow a lognormal process with volatility σ. It has one zero-coupon debt issue with face value D due at time T. Nothing happens before T.

At T, there are two outcomes. If V is greater than D, the debt is repaid and shareholders keep V − D. If V is less than D, shareholders walk away and lenders take V. So equity payoff is max(V − D, 0). That is a call option on the assets with strike D. This is why Black-Scholes-Merton machinery applies.

Lenders' payoff is D − max(D − V, 0). They hold risk-free debt and are short a put on the assets. So risky debt = D e^(−rT) − put. Equivalently, risky debt = V − equity. The put value rises with volatility and leverage, so risky debt falls in value and the credit spread widens.

The default probability comes from d2. Under risk-neutral valuation, P(V_T < D) = N(−d2). To get a real-world probability, replace r with the asset's expected return μ. The distance to default counts how many standard deviations the expected log asset value sits above the default point. A bigger distance means a lower default probability.

The KMV approach is the practical version. Asset value and volatility are backed out from observed equity and equity volatility. Distance to default is computed with a default point. This is then mapped to an expected default frequency (EDF) using a database of historical defaults, not the normal distribution. That mapping is the key difference from the pure Merton model.

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 (continuous), T = maturity.
d1 and d2
d1 = [ln(V ÷ D) + (r + σ²/2)·T] ÷ (σ·√T); d2 = d1 − σ·√T
σ is asset volatility, not equity volatility.
Risk-neutral default probability
P(default) = N(−d2)
Probability that V_T < D under risk-neutral measure.
Distance to default (real-world)
DD = [ln(V ÷ D) + (μ − σ²/2)·T] ÷ (σ·√T); P(default) = N(−DD)
Uses the expected asset return μ instead of r.
Value of risky debt
B = V − E = D·e^(−rT)·N(d2) + V·N(−d1) = D·e^(−rT) − Put
Debt is risk-free debt less a put on the assets.
Yield and credit spread
y = −(1 ÷ T)·ln(B ÷ D); spread = y − r
Continuous compounding. Spread rises with leverage and asset volatility.
Equity volatility link
σ_E·E = N(d1)·σ·V
Used to solve for unobservable V and σ from equity data.
KMV simple distance to default
DD = (V − default point) ÷ (σ·V); default point = short-term debt + 0.5 × long-term debt
EDF is read from an empirical mapping of DD to default rates.

How to solve Structural Models of Default (Merton Model) questions

Use this order for any Merton-model question. Keep all rates continuous and the horizon in years.

  1. 1Identify V, D (face value of the zero-coupon debt), σ (asset volatility), r and T. Check whether σ given is for assets or equity.
  2. 2If only equity data is given, solve for V and σ using the call formula and σ_E·E = N(d1)·σ·V.
  3. 3Compute d1 and then d2 = d1 − σ√T. Carry at least four decimals.
  4. 4Read N(d1) and N(d2) from the normal table or calculator. Use N(−x) = 1 − N(x).
  5. 5For default probability, use N(−d2). If the question gives an expected asset return μ, use μ in place of r.
  6. 6For equity use E = V·N(d1) − D·e^(−rT)·N(d2). For debt use B = V − E.
  7. 7For the spread, compute y = −ln(B ÷ D) ÷ T and subtract r.
  8. 8Sanity check: B must be below D·e^(−rT), E must be positive, and the spread must be positive.

Quickest way: Shortcut: d2 first, then work backwards

When to use it: Use when the question asks only for default probability, distance to default, or a qualitative change in spread.

  1. Compute d2 directly: [ln(V ÷ D) + (r − σ²/2)·T] ÷ (σ√T). This skips d1.
  2. Default probability = N(−d2). Keep N(−d2) small when d2 is above 2.
  3. For direction-of-change questions, reason first: higher σ, higher D, longer T for a weak firm, or lower V all raise default probability and spread.
  4. Eliminate options using bounds. Debt value must lie below D·e^(−rT), and the spread must be positive.
  5. On a financial calculator, use ln, e^x and the square root keys. Store σ√T in memory and reuse it.

Common mistakes in Structural Models of Default (Merton Model)

  • Using equity volatility as σ in d1 and d2

    Equity volatility is observable, so it feels like the natural input.

    Fix: The formulas need asset volatility. Equity volatility is larger because of leverage. Convert using σ_E·E = N(d1)·σ·V.

  • Using N(d2) as the default probability

    N(d2) is printed in the equity formula and looks like the answer.

    Fix: N(d2) is the probability of repayment. Default probability is N(−d2) = 1 − N(d2).

  • Mixing risk-neutral and real-world probabilities

    The formulas for d2 and DD look nearly the same.

    Fix: Use r for pricing and risk-neutral probabilities. Use μ for real-world distance to default.

  • Forgetting to discount the debt face value

    Students write V·N(d1) − D·N(d2).

    Fix: The strike is discounted: D·e^(−rT). Check the formula each time.

  • Computing the spread without taking the logarithm

    They use (D ÷ B − 1) ÷ T or forget to subtract r.

    Fix: Use y = −ln(B ÷ D) ÷ T for continuous compounding, then spread = y − r.

  • Assuming KMV EDF equals N(−DD)

    Both start from distance to default.

    Fix: KMV maps DD to EDF using empirical default data. The normal distribution is not used for that final step.

Worked examples

Example 1

A firm has assets of $120 million and a single zero-coupon debt with face value $100 million due in 1 year. Asset volatility is 20% and the risk-free rate is 5% (continuous). What is the risk-neutral probability of default? Options: A) 6.9% B) 14.4% C) 28.8% D) 85.6%

Show the solution
  1. Formula: P(default) = N(−d2), with d2 = [ln(V ÷ D) + (r − σ²/2)·T] ÷ (σ√T).
  2. ln(120 ÷ 100) = ln(1.2) = 0.1823.
  3. r − σ²/2 = 0.05 − 0.02 = 0.03, and T = 1, so the numerator = 0.1823 + 0.03 = 0.2123.
  4. σ√T = 0.20, so d2 = 0.2123 ÷ 0.20 = 1.0616.
  5. N(1.0616) ≈ 0.8558, so N(−1.0616) = 1 − 0.8558 ≈ 0.1442.

Answer: B) about 14.4%. Option D is N(d2), the repayment probability.

Example 2

A firm has assets of $200 million and zero-coupon debt of face value $150 million due in 1 year. Asset volatility is 30% and the risk-free rate is 5% (continuous). Given N(d1) = 0.8989 and N(d2) = 0.8354, find the credit spread on the debt (continuous compounding). Options: A) 0.95% B) 1.60% C) 2.31% D) 7.31%

Show the solution
  1. Equity: E = V·N(d1) − D·e^(−rT)·N(d2).
  2. V·N(d1) = 200 × 0.8989 = 179.78.
  3. D·e^(−0.05) = 150 × 0.951229 = 142.684, and × 0.8354 = 119.20.
  4. E = 179.78 − 119.20 = 60.58.
  5. Risky debt B = V − E = 200 − 60.58 = 139.42.
  6. Yield y = −ln(B ÷ D) = ln(150 ÷ 139.42) = ln(1.07588) = 0.0731, or 7.31%.
  7. Spread = 7.31% − 5.00% = 2.31%.

Answer: C) about 2.31% (231 basis points). Option D is the yield, not the spread.

Exam tips

  • Questions are often conceptual: what happens to equity, debt value, default probability or spread when σ, leverage or T changes. Learn the directions, not just the numbers.
  • Remember equity holders are long volatility (a call) and debt holders are short volatility (short a put). This explains incentive problems such as risk shifting.
  • Read carefully whether the question asks for risk-neutral or real-world probability. It changes whether you use r or μ.
  • Know the KMV steps: infer V and σ from equity, set the default point, compute DD, then map to EDF using empirical data.
  • Expect to be given N(d1) and N(d2) values. Spend your time on setting up the formula correctly and on avoiding the N(d2) versus N(−d2) trap.

Practice questions from Measuring Credit Risk

Structural Models of Default (Merton Model) 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 of Default (Merton Model): frequently asked questions

What is distance to default in the Merton model?

It is the number of standard deviations between the expected log asset value at the horizon and the default point (the debt face value). A larger distance means a lower default probability. In the pure Merton model, default probability is N(−DD).

How is the KMV EDF different from the Merton default probability?

KMV computes a distance to default using a default point of short-term debt plus half of long-term debt. It then maps that distance to an expected default frequency using historical default data. The Merton model instead uses the normal distribution directly.

How do you value risky debt with the Merton model?

Risky debt equals asset value minus equity, where equity is the call option value. Equivalently, it equals the present value of the face value, D·e^(−rT), minus a put option on the assets with strike D. The yield on this debt less r gives the credit spread.

Why does the credit spread rise when asset volatility rises?

Lenders are short a put on the firm's assets. Higher volatility makes that put worth more, so the risky debt is worth less. A lower debt value for the same face amount means a higher yield and a wider spread.