What’s the question?
A lender owns a claim that pays in full unless the company fails; a shareholder owns whatever is left afterwards. Merton (1974) turned that ordering into an option: equity is a call on the firm’s assets, struck at the face value of the debt. Cover the debt and shareholders keep the remainder; fall short and they hand the firm to its lenders.
The output is distance to default, the number of standard deviations of asset value separating a firm from its default point. It combines two things usually looked at separately: the debt on the balance sheet, and how violently the assets move in value. The question is whether the combination reorders the S&P 500 against a plain leverage screen, or reproduces it with extra arithmetic.
The approach
Asset value and asset volatility are not observable. Both are solved from the market value of equity, its volatility, and the reported debt.
- Take current S&P 500 members outside financials and real estate, where the assumption of a simple debt structure holds up. Companies without reported long-term debt leave the sample.
- Set the default point to debt due within one year plus half of longer-dated debt, the Moody’s KMV convention; long-dated debt need not be repaid at the horizon, so counting all of it overstates the barrier. Filers differ over what that line holds, so the noncurrent figure from the filing is used where a company reports it inclusive of the current portion, and a company reporting no noncurrent long-term debt for the quarter leaves the sample.
- Build a one-year daily path of equity value from the latest market capitalisation and the split-adjusted close, then solve E = V·N(d1) − F·e^(−rT)·N(d2) for asset value V at each date given a trial asset volatility, recompute asset volatility from the solved values, and repeat until the two agree. Convergence took at most eight passes.
- Read off distance to default at the last date, using the one-year Treasury yield of 3.77% on 5 August 2026 as the drift.
- Screen the panel first: at least 200 trading sessions, no session-to-session move beyond 45% in log terms, which marks a spin-off rather than a return, and a market capitalisation within 15% of shares times price.
Every company named below was checked against its own filings on the SEC’s XBRL interface.
Code
import numpy as np, pandas as pd, xfinlink as xfl
from scipy.stats import norm
xfl.set_api_key("YOUR_API_KEY") # free at https://xfinlink.com/signup
R, T = 0.0377, 1.0
def asset_value(E, F, sigma_V, r=R, t=T):
"""Invert E = V N(d1) - F exp(-rt) N(d2) for V, one V per equity observation."""
V = E + F
for _ in range(100):
sq = sigma_V * np.sqrt(t)
d1 = (np.log(V / F) + (r + 0.5 * sigma_V ** 2) * t) / sq
gap = V * norm.cdf(d1) - F * np.exp(-r * t) * norm.cdf(d1 - sq) - E
step = gap / np.maximum(norm.cdf(d1), 1e-8)
V = np.maximum(V - step, E * 1.000001)
if np.max(np.abs(step) / V) < 1e-12:
break
return V
def merton(E, F, r=R, t=T):
sigma_E = np.diff(np.log(E)).std(ddof=1) * np.sqrt(252)
sigma_V = sigma_E * E[-1] / (E[-1] + F) # KMV starting guess
for _ in range(300):
V = asset_value(E, F, sigma_V, r, t)
new = np.diff(np.log(V)).std(ddof=1) * np.sqrt(252)
if abs(new - sigma_V) < 1e-10:
sigma_V = new
break
sigma_V = new
V = asset_value(E, F, sigma_V, r, t)
dd = (np.log(V[-1] / F) + (r - 0.5 * sigma_V ** 2) * t) / (sigma_V * np.sqrt(t))
return V[-1], sigma_V, dd, norm.cdf(-dd)
members = xfl.index("sp500")["ticker"].dropna().tolist()
fun = xfl.fundamentals(members, period_type="quarterly", start="2025-09-01",
fields=["current_portion_long_term_debt", "long_term_debt"])
latest = fun.sort_values("period_end").groupby("ticker").tail(1).set_index("ticker")
latest["F"] = latest["current_portion_long_term_debt"].fillna(0) + 0.5 * latest["long_term_debt"]
mcap = xfl.metrics(list(latest.index), period_type="daily", fields=["market_cap"],
start="2026-07-27")
px = xfl.prices(list(latest.index), period="1y", fields=["adj_close"])
for tk, s in latest.iterrows():
path = px[px["ticker"] == tk].sort_values("date")["adj_close"].values
E0 = mcap[mcap["ticker"] == tk].sort_values("period_end")["market_cap"].iloc[-1]
V, sigma_V, dd, pdef = merton(E0 * path / path[-1], s["F"])
print(f"{tk:6} asset vol {sigma_V:5.2f} leverage {s['F'] / V:5.2f} DD {dd:6.2f}")Full script with formatting and visualisation: merton-distance-to-default-sp500-python.py
Output
============================================================================================================
MERTON DISTANCE TO DEFAULT | S&P 500 excluding financials and real estate | as at 2026-08-05
============================================================================================================
Equity as a one-year call on assets, strike = default point, risk-free rate 3.77% (1-year Treasury, 5 Aug 2026)
Default point = debt due within one year + half of longer-dated debt (KMV convention)
Asset value and asset volatility solved jointly from 1 year of daily equity values
S&P 500 members outside financials and real estate with reported debt 358
with market capitalisation, share count and closing price 345
with at least 200 trading sessions in the window 343
with no session-to-session move beyond 45% in log terms 340
market capitalisation within 15% of shares x price 340
Filing quarters used: 2025-12-31 to 2026-07-04
TWELVE SHORTEST DISTANCES TO DEFAULT
company equity default equity asset F/V DD implied leverage
$m point $m vol vol default rank
------------------------------------------------------------------------------------------------------------
CHTR CHARTER COMMUNICATIONS INC 18,399 47,479 0.50 0.17 0.74 1.88 3.00% 1
SMCI SUPER MICRO COMPUTER INC 18,234 2,330 0.92 0.82 0.11 2.28 1.13% 125
NCLH NORWEGIAN CRUISE LINE HLDGS LTD 9,321 8,088 0.54 0.30 0.47 2.48 0.65% 4
KMX CARMAX INC 8,042 8,798 0.58 0.25 0.53 2.58 0.49% 2
LITE LUMENTUM HOLDINGS INC 59,243 3,260 0.93 0.84 0.05 3.12 0.09% 226
NRG N R G ENERGY INC 25,390 12,384 0.48 0.35 0.33 3.12 0.09% 13
CZR CAESARS ENTERTAINMENT INC DE 6,131 5,958 0.49 0.22 0.50 3.19 0.07% 3
ORCL ORACLE CORP 415,843 61,171 0.64 0.58 0.13 3.28 0.05% 106
BLDR BUILDERS FIRSTSOURCE INC 8,162 2,302 0.50 0.41 0.22 3.55 0.02% 50
DOW DOW INC 21,384 9,420 0.44 0.31 0.31 3.75 0.01% 21
CCL CARNIVAL CORP 40,637 13,180 0.47 0.36 0.25 3.85 0.01% 40
VST VISTRA CORP 47,527 10,531 0.50 0.43 0.18 3.88 0.01% 68
TEN HEAVIEST DEBT LOADS, AND WHERE THE MODEL PUTS THEM
company sector F/V asset vol DD DD rank
------------------------------------------------------------------------------------------------------------
CHTR CHARTER COMMUNICATIONS INC Communication Services 0.74 0.17 1.88 1
KMX CARMAX INC Consumer Discretionary 0.53 0.25 2.58 4
CZR CAESARS ENTERTAINMENT INC DE Consumer Discretionary 0.50 0.22 3.19 7
NCLH NORWEGIAN CRUISE LINE HLDGS LTD Consumer Discretionary 0.47 0.30 2.48 3
PCG P G & E CORP Utilities 0.47 0.14 5.65 53
EIX EDISON INTERNATIONAL Utilities 0.46 0.14 5.83 56
ES EVERSOURCE ENERGY Utilities 0.37 0.16 6.29 78
EXC EXELON CORP Utilities 0.36 0.12 8.36 160
CAG CONAGRA BRANDS INC Consumer Staples 0.36 0.20 5.12 41
WYNN WYNN RESORTS LTD Consumer Discretionary 0.35 0.24 4.46 25
SECTOR MEDIANS
sector names F/V asset vol DD
------------------------------------------------------------
Information Technology 62 0.035 0.449 6.86
Communication Services 15 0.138 0.281 7.14
Materials 26 0.088 0.283 7.94
Consumer Staples 30 0.104 0.234 8.11
Consumer Discretionary 38 0.077 0.293 8.34
Energy 20 0.120 0.276 8.49
Health Care 50 0.092 0.283 8.77
Utilities 29 0.278 0.137 9.55
Industrials 70 0.062 0.278 10.53
Ranking agreement across 340 companies
Spearman, DD against leverage 0.514
Spearman, DD against asset volatility 0.448
Names in both the riskiest 20 by DD and by leverage 6 of 20
Riskiest by leverage only: CAG, CMCSA, CMS, CPB, DTE, DUK, EIX, ES, EXC, FE, PCG, PNW, VZ, WYNN
Riskiest by DD only: APTV, BLDR, CCL, DOW, INTC, KLAC, LITE, LYB, MOS, ON, ORCL, SMCI, UAL, VST
Distance to default: median 8.61, 5th percentile 4.07, 95th 16.48
SENSITIVITY OF THE RANKING
risk-free rate 2% mean |change| in DD 0.04 riskiest-20 held 20 of 20 Spearman 1.000
risk-free rate 6% mean |change| in DD 0.05 riskiest-20 held 20 of 20 Spearman 1.000
default point + short-term debt mean |change| in DD 0.40 riskiest-20 held 18 of 20 Spearman 0.983
6-month estimation window mean |change| in DD 0.99 riskiest-20 held 17 of 20 Spearman 0.962What this tells us
The two rankings disagree. Distance to default and leverage correlate at 0.514 across 340 companies, and only 6 of the 20 names flagged riskiest by one measure appear in the other’s twenty. Position on the plane fixes distance to default, so the grey curves on the chart are lines of equal credit risk, and they bend sharply: a company moves along one by trading debt for volatility. Charter Communications sits at the bottom right, the heaviest debt load in the sample at 74% of asset value against a cable network’s 17% asset volatility. Super Micro Computer sits at the top left, debt at 11% of assets and asset volatility at 82%. Both land inside a distance to default of 2.3.
Utilities make the point in aggregate. They carry the most debt, a median of 27.8% of asset value against 3.5% in information technology, and the model still ranks them second safest, because their median asset volatility of 13.7% is under a third of technology’s 44.9%.
The implied probabilities deserve less weight than the ordering. A risk-neutral 3.00% for Charter sits far above what its bond spreads imply, since the calculation prices default risk rather than forecasting it, and the model grants the firm one maturity and no refinancing. Bharath and Shumway (2008) found the ordering carries the predictive content, not the level.
The sensitivity checks say the ordering is stable. Moving the risk-free rate between 2% and 6% shifts distance to default by 0.05 on average and leaves the riskiest 20 untouched; adding short-term borrowings to the default point costs 0.40 and keeps 18. Halving the estimation window costs 0.99 and keeps 17, so the estimate carries about a point of noise from the volatility input alone.
So what?
Use the measure as a screen rather than a probability. A leverage sort cannot see how stable the business underneath is; this one does, and the disagreement list is where the work sits. Oracle, Intel and Vistra rank among the riskiest 20 by distance to default and nowhere near it by leverage, because asset volatility does the ranking there.
For a covenant or counterparty screen, read both coordinates rather than the single number: a firm at 0.74 leverage and 17% asset volatility needs refinancing capacity, one at 0.11 leverage and 82% volatility needs its equity to stop moving. A point of estimation noise also argues against reacting to small moves: a name drifting from 8.6 to 7.9 has said nothing, while one going from 6 to 3 has repriced. A guide to the Altman Z-Score covers an accounting-based distress score that can sit beside distance to default.
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