howard-lynn-ye Credit risk models
Two model families each produce a number called "the probability of default", and they are not
the same number. Merton's N(−d2) is risk-neutral; a rating-agency default table is
physical. Using one where the other belongs is the most expensive mistake in this domain, and
neither one carries a label.
Every figure below is printed by scripts/credit_models.py (runs in 1.8 s; QuantLib optional,
imported inside quantlib_cds_cross_check). ✅ Measured means this file produced it on 2026-09-09
with QuantLib 1.43, scipy 1.13.0, Python 3.11.3.
The rule:
N(−d2)is risk-neutral; a rating-table default rate is physical. Price with the first, forecast with the second, and never quote a hazard without its recovery.
1. ✅ Merton (1974), reproducing a textbook example
Equity is a call on the firm's assets struck at the face value of debt. Two equations pin the two unobservables:
E = V N(d1) - D e^{-rT} N(d2)
sigma_E = N(d1) sigma_V V / E
⚠️ The reference: Hull, Options, Futures, and Other Derivatives, the worked example in the credit-risk chapter on using equity prices to estimate default probabilities. Inputs: E = 3, sigma_E = 80%, D = 10 payable in 1 year, r = 5%. The book reports V = 12.40, sigma_V = 0.2123, N(−d2) = 12.7%, debt worth 9.40 against a promised PV of 9.51, and an expected loss of about 1.2%. Those published values are secondhand; everything in the "solved" column is ✅ measured here.
| quantity | ⚠️ Hull | ✅ solved here |
|---|---|---|
| asset value V | 12.40 | 12.395387 |
| asset volatility sigma_V | 0.2123 | 0.212305 |
| d2 | 1.1408 | 1.140826 |
| risk-neutral PD = N(−d2) | 12.7% | 12.6971% |
| market value of debt | 9.40 | 9.395387 |
| PV of the promised payment | 9.51 | 9.512294 |
| expected loss | about 1.2% | 1.2290% |
| implied credit spread | — | 123.7 bp |
✅ Solve residual 1.1e-16, and pushing (V, sigma_V) back through the forward map returns equity 3.0000000000 and equity vol 0.8000000000. The reproduction is exact to every digit the book prints.
1b. ✅ The solver fails loudly, which is the property that matters
A two-equation nonlinear solve can converge to a plausible wrong root, and that is the only failure mode you would not notice. ✅ Measured over a 270-point grid (E ∈ {0.1 … 50}, sigma_E ∈ {20% … 300%}, D ∈ {1, 10, 100}, T ∈ {0.25, 1, 5}):
- 263 solved and round-tripped through the forward map to 1e-8
- 7 raised — all at sigma_E = 300% with T = 5 (an equity vol that high over five years is a firm the two equations cannot separate)
- 🚨 0 returned a wrong root
Keep that property if you rewrite this. Start from V = E + D e^{−rT} and
sigma_V = sigma_E·E/V, and check the residual of both equations before you use the answer.
⚠️ The second equation is an instantaneous identity, not a law. sigma_E = N(d1)·sigma_V·V/E
holds at a point in time; equity vol is not constant as V moves, so an equity vol estimated over a
year and an instantaneous relation are being married by assumption. KMV-style implementations
iterate on a time series of V instead. This skill implements the textbook two-equation form, which
is what the reproduction above is of.
2. 🚨 The same firm, two default probabilities
d2 carries the drift. Under the risk-neutral measure the drift is r; the physical
distance-to-default uses the asset drift mu. ✅ Measured on the Merton firm from §1:
| asset drift mu | distance to default | PD physical | PD risk-neutral | ratio |
|---|---|---|---|---|
| 5.00% (= r) | 1.1408 | 12.697% | 12.697% | 1.00 |
| 8.00% | 1.2821 | 9.990% | 12.697% | 1.27 |
| 10.00% | 1.3763 | 8.436% | 12.697% | 1.51 |
| 15.00% | 1.6118 | 5.350% | 12.697% | 2.37 |
✅ mu = r reproduces N(−d2) exactly — that is the check that the two formulas are the same formula. Every mu > r gives a smaller physical PD, and at a 15% asset drift the risk-neutral probability is 2.37× the physical one. The gap is the credit risk premium; it is not an error in either number.
3. Constant hazard and the exact CDS — and what lambda(1 − R) costs
Survival is exp(−lambda·t). ✅ Measured, lambda = 2%, R = 40%, r = 3%, 5-year CDS, quarterly
premiums in arrears with accrual on default — everything in closed form, nothing discretised:
| leg | value per unit notional |
|---|---|
| survival to 5y | 0.904837 |
protection leg (1−R)·lambda/(r+lambda)·(1 − e^{−(r+lambda)T}) |
0.05308781 |
| premium annuity | 4.39639204 |
| + accrual on default | 0.01103692 |
| = risky PV01 | 4.40742896 |
| exact par spread | 120.4507 bp |
lambda(1 − R) |
120.0000 bp — error −0.4507 bp |
| exact par spread without accrual on default | 120.7531 bp (+0.3024 bp) |
✅ The approximation error is a constant fraction of the spread, and it is bigger than you would guess:
| lambda | 0.5% | 1.0% | 2.0% | 5.0% | 10.0% | 20.0% |
|---|---|---|---|---|---|---|
| exact (bp) | 30.11 | 60.23 | 120.45 | 301.13 | 602.25 | 1204.47 |
lambda(1−R) (bp) |
30.00 | 60.00 | 120.00 | 300.00 | 600.00 | 1200.00 |
| error (bp) | −0.11 | −0.23 | −0.45 | −1.13 | −2.25 | −4.47 |
| error (%) | −0.37 | −0.37 | −0.37 | −0.37 | −0.37 | −0.37 |
The approximation UNDERSTATES the spread, by exactly 0.37% at every hazard. A constant relative error is a clue, and it has a clean explanation.
✅ lambda(1 − R) is not an approximation at all — it is the continuous-premium spread
With a continuously-paid premium the annuity is (1 − e^{−cT})/c with c = r + lambda, and the
protection leg is (1−R)·lambda/c·(1 − e^{−cT}). The ratio is (1−R)·lambda exactly — r,
T and the exponentials all cancel. ✅ Reproduced to 12 digits.
🚨 So the entire gap is the premium FREQUENCY, not the hazard model. Paying in arrears defers the annuity, which raises the fair spread. ✅ Measured at lambda = 2%, R = 40%, r = 3%, T = 5:
| premiums per year | 1 | 2 | 4 (market) | 12 | 52 | 365 |
|---|---|---|---|---|---|---|
| exact (bp) | 121.8120 | 120.9030 | 120.4507 | 120.1501 | 120.0346 | 120.0049 |
| without accrual on default (bp) | 123.0506 | 121.5126 | 120.7531 | 120.2503 | 120.0577 | 120.0082 |
vs lambda(1−R) |
+1.8120 | +0.9030 | +0.4507 | +0.1501 | +0.0346 | +0.0049 |
It converges to 120.0000 bp as the premium goes continuous. Accrual on default gives part of
that annuity back (−0.30 bp at quarterly, −1.24 bp at annual), which is why the two columns
converge from opposite sides. Use lambda(1−R) to sanity-check an order of magnitude; the
0.45 bp it misses at quarterly is real money on size, and 1.81 bp on an annual-pay contract is
not a rounding error.
4. 🚨 Recovery is an input, and 40% is a convention
The market quotes a spread. The hazard you back out of it depends entirely on the recovery you assumed. ✅ Measured, market 5-year spread 100 bp, r = 3%:
| recovery R | implied lambda (exact) | s/(1−R) |
5-year PD |
|---|---|---|---|
| 0% | 0.9963% | 1.0000% | 4.859% |
| 20% | 1.2453% | 1.2500% | 6.037% |
| 40% | 1.6604% | 1.6667% | 7.967% |
| 60% | 2.4906% | 2.5000% | 11.709% |
| 80% | 4.9813% | 5.0000% | 22.047% |
🚨 The same 100 bp quote implies a hazard 2.00× higher at R = 60% than at R = 20%, and a 5-year default probability of 22.0% instead of 6.0% at R = 80%. A hazard rate without its recovery assumption is not a number. The standard 40% is a convention inherited from the CDS market, not a measurement of anything about the issuer.
The saving grace: spread and recovery move together in the pricing, so the price of a protection leg is much less sensitive than the hazard is. That is why quoting (lambda, R) as a pair matters more than getting R right.
5. 🚨 The trap: pricing with a historical default probability
Rating-agency and internal default tables are physical. CDS spreads, bond prices, and CVA all need the risk-neutral probability. ✅ Measured on the same Merton firm at an asset drift of mu = 10%:
| physical | risk-neutral | |
|---|---|---|
| 1-year PD | 8.436% | 12.697% |
| implied hazard | 8.813% | 13.579% |
| 1-year CDS fair spread (R = 40%) | 532.1 bp | 819.8 bp |
🚨 −287.7 bp. You would sell protection at 532 bp on a name whose fair spread is 820 bp, and your model would show a profit.
🚨 And then you would mark it. Revaluing the fair risk-neutral contract with the physical hazard reports a value of −2.67% of notional that does not exist. The position looks mispriced against you, so the natural reaction is to add more of it.
The direction is systematic, not random. Physical PD < risk-neutral PD for any firm with a positive risk premium (§2), so this trap always makes credit look cheap and always makes you short protection.
6. ✅ Cross-check against QuantLib 1.43
Same flat hazard, flat rate, 5-year quarterly schedule:
| spread | vs this file | |
|---|---|---|
FlatHazardRate.survivalProbability(5) |
0.904837 | matches exp(−0.02·5) |
MidPointCdsEngine.fairSpread() |
120.4522 bp | mine 120.4507, +0.0015 bp |
IsdaCdsEngine.fairSpread() |
120.4368 bp | −0.0139 bp |
MidPointCdsEngine, settlesAccrual=False |
120.7544 bp | mine 120.7531, +0.0013 bp |
✅ Agreement to about a hundredth of a basis point, and the residual is explained rather than waved at: the engines accrue coupons on ACT/365 dates and place the default time at the midpoint of each quarter, against the exact integral computed here. ⚠️ Note the two QuantLib engines differ from each other by 0.0154 bp on identical inputs — the ISDA model has its own conventions. If you are reconciling to a counterparty, the engine is part of the trade description.
7. What the script gives you
scripts/credit_models.py — scipy only at import (brentq, fsolve, norm).
| Function | Does |
|---|---|
merton_solve(E, sigma_E, D, r, T) |
§1, returns a frozen MertonResult with residual |
merton_equity(V, sigma_V, D, r, T) |
the forward map — the round-trip check |
merton_solver_scan() |
§1b, the 270-point robustness grid |
distance_to_default(V, sigma_V, D, mu, T) |
§2, physical DD and N(−DD) |
survival_probability / hazard_from_default_probability |
the constant-hazard pair |
cds_legs(hazard, recovery, r, T, freq, accrual_on_default) |
§3, every leg in closed form |
cds_par_spread / implied_hazard / cds_mark_to_market |
price, invert, revalue |
approximation_error_table |
§3, what lambda(1−R) costs |
recovery_sensitivity(spread, recoveries, r, T) |
§4 |
measure_trap(...) |
§5, the same firm priced both ways |
quantlib_cds_cross_check(...) |
§6, or None |
Where this sits
../term-structure-models/SKILL.md— the discount curve every leg above is priced on, and the day-count and compounding conventions that shift a 5-year rate by 17.68 bp. A CDS spread quoted against the wrong curve convention is wrong before the credit model starts.../../../fin-libraries/skills/lib-quantlib/SKILL.md— 🚨Settings.instance().evaluationDateis a global and a stale one gives NPV exactly 0.0; §6 sets it before building any curve. AlsoDefaultProbabilityTermStructureHandle,FlatHazardRate, and the CDS engines.../../../fin-core/skills/derivatives-pricing/SKILL.md— 🚨rateslibis not open source andfinancepyis GPL-3.0; QuantLib is the permissive route for CDS.../../../fin-core/skills/portfolio-and-risk/SKILL.md— portfolio credit, correlation and capital. This skill is one name at a time.../../../fin-core/skills/fundamental-and-macro-data/SKILL.md— where the equity value, the debt face value and the spread quotes come from, and their point-in-time behaviour. A Merton PD computed from a restated balance sheet is a backtest artefact.../option-pricing-models/SKILL.md— Merton's equity call is a Black-Scholes option, so everything there aboutN(d1),N(d2)and dividend yields applies to the firm's assets too.