Credit & distress

We solve two equations at once for the firm's implied asset value V and asset volatility σ_A — treating equity as a call option on the firm's assets struck at the debt barrier D (KMV default point: short-term debt + half of long-term debt). The 252-day realized equity volatility σ_E is an input to that solve — it enters the second pricing equation (the equity-vol to asset-vol Itô link) and is what pins down σ_A; it does not appear in the DTD formula itself. With the solved values, DTD = (ln(V/D) + (μ − ½σ_A²)·T) / (σ_A·√T), with T = 1 year. The drift μ = max(r, trailing 1-year equity return), capped at 50% (Bharath-Shumway naive form) — note the risk-free floor: any name whose 1-year return fell below r (most names in a down year) gets μ = r, not the negative equity return. The result is the number of asset-volatility standard deviations the firm stands above default. Clamped to ±25σ.

A falling DTD (especially below 2) tightens the case that structural stress is building — set it next to the accounting gauges (Altman Z, forensic flags) to triangulate. A high DTD (5+) weakens a distress narrative but says nothing about valuation or earnings quality. Because it moves with market price, it can compress fast in a sell-off even when fundamentals are unchanged — weigh that when judging how much signal it adds versus noise on volatile names.

Above ~4σ: the firm's asset cushion is wide — structural default within one year is remote. Around 2–4σ: meaningful but not acute pressure; worth cross-checking debt maturities. Below 2σ: thin buffer; the asset base is within two bad swings of the barrier. Below 1σ or negative: market-implied asset value is near or below the debt barrier — an acute structural-stress signal, consistent with elevated credit spreads or distressed trading levels.

Clamped at ±25σ in code. Near-debtless firms can reach 15–25σ (numerically, not practically meaningful beyond ~8). Healthy large-caps typically land 4–10σ. Stressed firms sit 1–3σ. Sub-1σ or negative values mark acute distress. The companion PD = N(−DTD) maps this to a 1-year probability: DTD of 4σ → PD ≈ 0.003%, DTD of 1σ → PD ≈ 16%.

PD = N(−DTD) is a Gaussian map, not an empirical default-frequency table — it understates true default risk in the 1–3σ range, where observed bankruptcy rates run above the bell-curve prediction. Treat the absolute PD number as directional, not actuarial.

We solve the Merton model iteratively: treating equity as a call option on the firm's assets, we back out the unobservable asset value V and asset volatility sigma_A from today's market cap and trailing realized equity vol. Two equations are solved together by fsolve (merton.py:344-358) — equity as a risk-neutral call on assets (priced off the risk-free rate r), plus the Ito link tying equity vol to asset vol. Distance-to-default (DTD) measures how many asset-vol swings stand between the firm and the KMV default barrier (short-term debt + half of long-term debt), using a physical drift mu in the numerator (the Bharath-Shumway "naive" variant); the result is clamped to plus or minus 25 sigma (merton.py:370). PD = N(-DTD), the Gaussian tail probability (merton.py:375). Inputs: market cap from intel_ticker_snapshots, 252-day realized vol from intel_daily_bars (monthly-panel fallback), and the debt barrier from point-in-time balance-sheet filings (intel_fundamentals_pit). Run nightly by backend/scripts/refresh_merton.py via llm_lab/merton.py; result stored in merton_credit_snapshots.prob_default.

Treat it as a structural floor check, not a verdict. A low reading removes one specific worry — the model says assets currently sit far above the debt barrier. A higher reading flags that the margin is thin enough to warrant digging into the debt schedule, upcoming maturities, and whether equity vol is elevated for an idiosyncratic or macro reason. Pair it with Altman Z and the debt ladder rather than reading it alone. Keep in mind the displayed number is the raw Gaussian PD, which (see the catch) does not track observed default frequencies closely — so use it to rank firms and spot thin cushions, not as a literal one-year default odds.

0 to 1 (0% to 100%) on the displayed Gaussian scale. Typical solvent large/mid-cap: under 0.5%. Elevated: 1-5%. Distress zone: above 10%. For contrast, the pooled empirical default rates from dd_edf.json (ex-financials, 1-year bankruptcy/delist outcome): the lowest DTD bucket (below -0.65 sigma) defaults ~25% of the time (bucket-0 EDF = 0.255), versus a Gaussian PD of 88% at that point. The empirical rate decays slowly with distance and stays around 0.25% out at DTD ~5 sigma, only dropping under 0.1% past roughly 6.7 sigma — far flatter than the Gaussian, which has already gone to near-zero by ~4 sigma.

The live number uses N(-DTD) — a Gaussian formula — not the empirical default-frequency table, so it is miscalibrated in both tails. In the thin-cushion region (DTD below ~2 sigma) the Gaussian overstates true default rates by roughly 3-7x (e.g. N(-DTD)=88% vs observed ~25% at DTD ~-0.65 sigma; peak ~6.6x in the sub-2-sigma buckets). The larger error is the opposite tail: above ~4 sigma the Gaussian drives PD to essentially zero while the empirical EDF stays around 0.1-0.6% (edf/gaussian ~98x at ~4 sigma and exploding beyond), so high-DTD names show a falsely perfect 0%. The empirical EDF calibration (dd_edf.json, AUC 0.91, built 2026-06-10) exists but is not yet the live path (pd_method = 'gaussian_dtd' in merton_credit_snapshots). Banks and financials are excluded entirely (merton_construct_invalid) — the model is construct-invalid for deposit-funded balance sheets.

Five ratios — working capital, retained earnings, and EBIT (each divided by total assets), market cap divided by total liabilities, and revenue divided by total assets — weighted 1.2/1.4/3.3/0.6/1.0 and summed. When EBIT isn't available the engine falls back to operating income for that term. Inputs come from the latest annual period in our fundamentals store; if any of the five is missing, the score is withheld rather than estimated. For financials (banks, insurers, brokers), real estate, and utilities the score is suppressed entirely (sector gate in segment_gating.py) because their balance sheets make the inputs meaningless; an unknown sector is not suppressed.

A Grey or Distress reading raises the bar for any bullish thesis — you'd want to see why liabilities are high relative to assets and whether cash generation is improving before treating upside estimates as credible. A Safe reading removes solvency as an objection but adds nothing positive on its own; pair it with the Piotroski F-Score to check whether operating quality backs the balance-sheet picture.

Above 2.99 puts the company in the Safe zone — low near-term bankruptcy risk by the model's standards. Between 1.81 and 2.99 is the Grey zone, where distress is possible but not certain. Below 1.81 is the Distress zone, historically associated with elevated default rates within two years.

No hard floor or ceiling, but the practical range is roughly −1 to 10+. Healthy large-caps typically score 3–6. The meaningful thresholds are 1.81 (Distress cutoff) and 2.99 (Safe cutoff); scores well above 3 add diminishing information. Negative scores occur when retained earnings are deeply negative (accumulated losses exceed assets).

The 1968 weights were fit on U.S. manufacturers and have never been recalibrated. A capital-light software company with high retained-earnings deficits (common in growth names) can land in Distress while being nowhere near bankruptcy; always check whether the company type makes the formula applicable.