4.4.3 Distance to Default & Default Probability

Basic Prerequisites: 02 · The Merton Structural Model.

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1. Intuition & Practical Objective

Once you have inverted equity into firm value and asset volatility, default probability is one step away - but the step contains the field’s most important distinction. Distance to default (DD) is the number of asset-volatility standard deviations between today’s firm value and the default point:

$$\mathrm{DD}=\frac{\ln (V/D)+(\mu -\frac{1}{2}{\sigma }_V^2)T}{{\sigma }_V\sqrt{T}}.$$

It answers the practitioner’s question directly: how far is this firm from the wall, measured in the units in which the firm actually moves? Two firms with the same leverage are not equally risky - the one with the more volatile assets is closer to default in the only metric that matters.

The subtlety: which drift $\mu$? Use the risk-neutral drift $r$ and $\mathrm{PD}=N(-\mathrm{DD})$ is the risk-neutral default probability $N(-d_2)$ - the right object for pricing (CDS, bonds, CVA). Use the physical (expected) return $\mu$ and you get the real-world PD - the right object for risk management and capital. These differ, and the risk-neutral PD is systematically higher. Getting this distinction wrong is the most common conceptual error in credit risk.

Why DD is the right coordinate. $N(-\mathrm{DD})$ is monotone in DD, and DD separates the two things that matter - leverage ($\ln (V/D)$) and business risk (${\sigma }_V$) - while normalising by the horizon. KMV’s empirical EDF mapping is built entirely on DD.

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2. Mathematical Ground Truth & Derivations

2.1 Real-world vs risk-neutral default probability

Under the physical measure the firm value has drift $\mu$; under the risk-neutral measure it has drift $r$. The probability the firm defaults is $\mathbb{P}(V_T<D)$ in each world, and both are $N(-\text{DD})$ with the appropriate drift:

$${\mathrm{DD}}^{\text{RN}}=\frac{\ln (V/D)+(r-\frac{1}{2}{\sigma }_V^2)T}{{\sigma }_V\sqrt{T}}=d_2,{\mathrm{DD}}^{\mathbb{P}}=\frac{\ln (V/D)+(\mu -\frac{1}{2}{\sigma }_V^2)T}{{\sigma }_V\sqrt{T}}.$$

So $\mathbb{Q}(\text{default})=N(-d_2)$ exactly, while the real-world $\mathrm{PD}=N(-{\mathrm{DD}}^{\mathbb{P}})$. Because risky assets have a risk premium, $\mu >r$, hence ${\mathrm{DD}}^{\mathbb{P}}>{\mathrm{DD}}^{\text{RN}}$ and

$$\underset{\text{risk-neutral}}{\underbrace{N(-d_2)}}>\underset{\text{real-world}}{\underbrace{N(-{\mathrm{DD}}^{\mathbb{P}})}}\text{for a risk-averse market.}$$

(Hull §24.5: risk-neutral PDs exceed historical PDs; the gap is the risk premium. The real-world PD is the one to compare with S&P/Moody’s historical default tables.)

2.2 The KMV default point (Crosbie–Bohn empirical refinement)

Plain Merton sets the barrier at total debt $D$. Moody’s KMV found empirically that firms default when assets fall below

$$D^{\ast }=\text{short-term debt}+\frac{1}{2}\text{long-term debt},$$

because long-term debt does not come due immediately. Substituting $D^{\ast }$ for $D$ in $\ln (V/D)$ raises DD for long-dated-debt-heavy firms and is the practical form used by EDF vendors. (Bluhm §1.2.3 refers to this as the calibrated default point of the asset-value model.)

2.3 From PD to credit spread (Merton eq. 14)

The spread is not $N(-d_2)$ scaled - it is the risky yield minus $r$ obtained from the debt value:

$$R(t)-r=-\frac{1}{t}\ln \{\Phi [h_2]+\frac{1}{d}\Phi [h_1]\},d=D{e}^{-rt}/V,R(t)=-\frac{1}{t}\ln \frac{F}{D}.$$

For a firm at the money (large ${\sigma }_V$, high leverage) this is the price of the put the shareholders hold; for a safe firm it collapses to a fraction of a basis point - which is exactly where the model fails empirically (page 05).

2.4 The DD → PD map in practice

Three mappings coexist: (i) the Gaussian $N(-\mathrm{DD})$; (ii) KMV’s empirical EDF map, which maps DD to an observed historical default rate (fatter-tailed than Gaussian); and (iii) rating-agency cumulative default tables. All three are monotone in DD; only (i) is the model’s own output.

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3. Computational Implementation - DD, both PDs, and the credit spread

Continuing the worked firm from page 02 ($E_0=100$, ${\sigma }_E=0.40$, $D=350$, $T=1$, $r=5\%$). Stdlib only.

Three verified takeaways: (1) the risk-neutral PD is $26.4$ bp; (2) a realistic risk premium ($\mu =10\%$) lowers the real-world PD to $4.3$ bp - six times smaller; (3) the model’s own credit spread is $0.71$ bp, which no market would ever quote for this firm. (2) and (3) are the seeds of page 05.

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4. Failure Modes & First-Principles Breakdowns

1. Risk-neutral ≠ real-world PD. Quoting $N(-d_2)$ as “the probability of default” for risk purposes is wrong by a factor of several here. Pricing uses $\mathbb{Q}$; capital and limits use $\mathbb{P}$ (Hull §24.5).
2. The Gaussian PD is too thin-tailed. $N(-\mathrm{DD})$ underestimates tail default rates; KMV re-maps DD onto empirical EDFs for this reason. The $N(-d_2)$ from a lognormal firm value is a model probability, not a robust frequency (see EVT & Fat Tails).
3. DD is only as good as the inversion. DD inherits every error from the $(V,{\sigma }_V)$ solve. In practice ${\sigma }_V$ is estimated with large noise, and DD is highly sensitive to it (page 05 quantifies a $\sim 9\%$ PD move per $1\%$ move in ${\sigma }_E$).
4. The default point is a judgement call. $D$, or $D^{\ast }=\text{ST}+\frac{1}{2}\text{LT}$, or a barrier - the DD level shifts with the choice, and it is calibrated, not derived.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Merton (1974)
• Bluhm, Overbeck & Wagner - Introduction to Credit Risk Modeling
• Gupton, Finger & Bhatia - CreditMetrics™ Technical Document (1997): the transition-matrix analogue (PD read off the default column) as the discrete alternative to DD.

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6. Connected Graph Bridges

• Back: 02 · Structural Model
• Forward: 04 · Reduced-Form & CDS · 05 · Failure Modes · Index Hub
• Sibling: VaR & Expected Shortfall (the quantile of the credit loss distribution DD feeds)