Common mistake 1 with Merton distance to default

Reading the risk-neutral PD as the actual expected default rate. It is systematically too high because it omits the asset risk premium.

Common mistake 2 with Merton distance to default

Trusting the level of PD for a single firm. The model's strength is the RANKING; the absolute number is sensitive to the debt figure and the horizon.

Common mistake 3 with Merton distance to default

Applying it to a firm with a complex or rapidly changing capital structure, where "one debt, one maturity" is a poor summary.

Common mistake 4 with Merton distance to default

Applying it to a bank. The model treats assets as observable and lognormal, and a leveraged financial institution violates both badly enough that the distance to default is not comparable with an industrial firm's.

EconStatsCraft › Merton distance to default

Merton distance to default

Treats equity as a call option on the firm and backs out the unobserved asset value and asset volatility, then reports how many standard deviations of cushion sit above the debt and the implied probability of default.

The intuition

Equity in a leveraged firm is a call option on the firm's assets, struck at the face value of its debt. If the assets are worth more than the debt at maturity, shareholders repay and keep the difference; if not, they hand over the firm and walk away, because limited liability caps their loss at zero. Once you see that, the whole option-pricing machinery becomes a credit model, and default probability is just the chance the option finishes out of the money.

When to use it

Estimating a firm's probability of default from market data, ranking credits, or understanding why default risk explodes as leverage rises. It underlies commercial scores such as Moody's KMV EDF.

Where the estimator comes from

Let asset value follow dV=\mu V\,dt+\sigma_V V\,dW and let debt be a zero-coupon claim of face D maturing at T. Equity pays \max(V_T-D,0), which is a call, so E=V\Phi(d_1)-De^{-rT}\Phi(d_2) with d_1=[\ln(V/D)+(r+\sigma_V^2/2)T]/(\sigma_V\sqrt{T}). DISTANCE TO DEFAULT is d_2: the number of standard deviations of asset value between the firm and its default point. The risk-neutral default probability is \Phi(-d_2). In practice V and \sigma_V are unobservable, so they are backed out by solving the equity equation jointly with \sigma_E E=\Phi(d_1)\sigma_V V, which comes from Ito applied to E(V).

The model

E = V\,N(d_1) - F e^{-rT} N(d_2),\qquad \sigma_E E = N(d_1)\,\sigma_V V,\qquad \text{DD} = \frac{\ln(V/F) + (\mu - \tfrac12\sigma_V^2)T}{\sigma_V\sqrt{T}},\qquad \text{PD} = N(-\text{DD})

What it assumes

Assets follow a geometric Brownian motion with constant volatility.
One zero-coupon debt with a single maturity - real capital structures are collapsed to one number F.
Default can only happen at the horizon T, not before (no early trigger from covenants).
Markets are efficient enough that the equity price and its volatility carry the information about the assets.
The firm's debt can be treated as a single zero-coupon bond maturing at a known date. Real capital structures have several maturities and seniorities, and collapsing them into one face value is the model's largest simplification.
Asset value follows a lognormal diffusion with no jumps, so default is always preceded by a gradual decline. Defaults that arrive suddenly - fraud, a lost lawsuit - are outside the model by construction.
Equity and its volatility are measured well enough to invert. For a thinly traded or recently listed firm the implied asset value inherits every error in the equity volatility estimate.
Default can only occur at the horizon, not before it. A firm that breaches its covenants after six months and is wound up does not appear in this model at all.

Statistical properties

Distance to default is a standardised measure and is comparable across firms; the raw default probability is not, because it depends on the assumed distribution.
Phi(-d2) is a RISK-NEUTRAL probability. It systematically exceeds the real-world default frequency, often by a wide margin, because it embeds a risk premium. Moody's KMV maps it to an empirical frequency rather than using it directly.
The model assumes default can only happen at maturity T. First-passage extensions (Black and Cox, 1976) allow default the moment assets cross a barrier, which raises the probability.
It is highly sensitive to asset volatility, which is the input that must be estimated rather than observed.
Because it is driven by market equity prices, it updates daily and leads agency ratings, which is its main practical advantage.
It systematically understates short-horizon default risk for firms far from the barrier, because diffusion cannot jump. Adding jumps or a stochastic barrier repairs this.

How to read the output

Asset volatility σ_V is always below equity volatility σ_E - leverage levers the equity's risk up.
Distance to default is the headline: how many standard deviations of asset value sit between the firm and the point where assets equal debt. Roughly, DD ≈ 4 is investment grade, DD ≈ 1–2 is distressed.
The probability of default reported here uses the risk-free drift, so it is risk-neutral - an upper bound on the true chance. Swap in an expected asset return to get a real-world PD.
The PD-versus-leverage curve is flat then steep: past roughly 70% leverage a little more debt moves the default probability a lot.
Distance to default is the output worth using: it is in standard deviations and is comparable across firms, while the probability attached to it depends entirely on the lognormal assumption.
Asset volatility is always below equity volatility, and the ratio between them is the leverage. If the model returns an asset volatility close to the equity volatility, the firm is barely levered or the inputs are wrong.

A worked reading

The demo solves the problem the way a credit analyst actually meets it: equity is observable and assets are not. Given equity of 100 with a volatility of 45%, debt with a face value of 90, one year and a risk-free rate of 4%, the model backs out an asset value of 186.4650 and an asset volatility of 24.145%. Note that asset volatility is far below equity volatility - always, because equity is a levered claim on the assets, and here leverage of 48.27% turns 24% asset volatility into 45% equity volatility. Distance to default comes out at 3.0619 standard deviations, giving a risk-neutral default probability of 0.110%. Treat that as an ordering statistic rather than a forecast. Real-world default frequencies at three standard deviations of distance are higher than a tenth of a percent, and the gap is the credit risk premium plus the fact that real asset returns are not lognormal - not an error in the arithmetic. The model is at its best ranking firms against each other and at its worst quoting an absolute probability.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Default probability essentially zero	Distance to default above 4, or volatility understated	Check the asset volatility input
Distance to default negative	Assets already below the debt face value	The firm is technically insolvent on this measure
Probability far above the observed default rate	Risk-neutral, not real-world	Expected. Map it empirically before quoting it
Very unstable week to week	Equity volatility moving the solved asset volatility	Smooth the volatility input over a longer window
Solver fails to converge	Poor starting values for V and sigma-V	Start from V = equity + debt and sigma-V = sigma-E times E over V
A firm with no traded equity	The model needs a market price	It cannot be applied. Use an accounting-ratio model

Common mistakes

Reading the risk-neutral PD as the actual expected default rate. It is systematically too high because it omits the asset risk premium.
Trusting the level of PD for a single firm. The model's strength is the RANKING; the absolute number is sensitive to the debt figure and the horizon.
Applying it to a firm with a complex or rapidly changing capital structure, where "one debt, one maturity" is a poor summary.
Applying it to a bank. The model treats assets as observable and lognormal, and a leveraged financial institution violates both badly enough that the distance to default is not comparable with an industrial firm's.

Where to read this properly

Merton (1974), "On the Pricing of Corporate Debt", Journal of Finance 29(2).
Black and Cox (1976), "Valuing Corporate Securities", Journal of Finance 31(2) - the first-passage version.
Crosbie and Bohn (2003), Modeling Default Risk, Moody's KMV - how the mapping is done in practice.
Duffie and Singleton, Credit Risk (2003), ch. 3.

Related Finance & risk

Black–Scholes option pricing · Bond price, duration & convexity · EMI / home loan planner · CAPM / beta estimation · Multi-factor model · Portfolio optimization · Value at Risk & Expected Shortfall · Performance & drawdown · Time value of money & NPV/IRR · Equity valuation (DDM, FCF, residual income) · Cost of capital (WACC) · Binomial option pricing · Option strategies (payoff & profit) · Forwards, futures & swaps · Credit risk & Basel capital · Extreme value theory (tail risk)

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Merton distance to default

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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