TiC Rating.

Agency ratings are careful, slow, and cover whom they choose to cover. When the U.S. Congress asked in 2012 whether rating systems could be standardised, the SEC's report concluded it may not be feasible given how differently each system is built · SEC 2012, quoted on deck s79. Banks meanwhile sit on a harder problem: for a single large company there is no default frequency to count. There is exactly one IBM — it either fails or it does not, and a probability estimated on a sample of one is not an estimate.

The chart below is the sharper surprise. Take every issuer Moody's rates B — the same letter, the same meaning, supposedly — and count the share that defaulted within a year. The answer swings from 0.0% to 9.8% depending on the year you ask.

The orthodox reading is that this is a flaw. The method's founding observation is that it is not: the default rate rides the credit cycle; the letter holds the company's place in the queue. A rating encodes the shape of a company's whole survival distribution, and the one-year PD is just the slice of it the current cycle exposes. Stock prices watch that distribution every trading day — which suggests computing the rating from the market instead of waiting for one. The rest of this page is how.

Three quantities describe a levered company, and they are awkward in three different ways: the asset value is a market value nobody quotes; the debt is a promise due in the future; only the equity trades today · deck s55. Merton's 1974 observation connects them: at the debt's maturity, shareholders keep whatever exceeds the debt and walk away from anything less. That payoff — everything above a strike, nothing below — is a call option's.

The identity matters because it points the telescope the useful way round: the market prices the option (equity) all day long, and option prices can be inverted for the value of the thing underneath. The market cap on this site is the day's dividend-adjusted close times that quarter's basic share count — so a 2017 equity value uses 2017's float, and dividends paid out are added back rather than read as losses.

Under Black–Scholes–Merton the equity of a levered firm is priced as a call on its assets:

Divide through by the discounted debt and the equation collapses to one shape. Both sides become ratios — x, assets over discounted debt, and z, equity over discounted debt — and the whole pricing relation is a single function g:

g rises monotonically in x: more assets always mean more equity. A monotone function is invertible by the oldest trick there is — bisection. Each trading day the engine observes z, walks the bracket down to the unique x with g(x) = z, and books that day's asset value. No gradient, no guess, no dependence on a starting point.

The inversion needs σ_A — but σ_A is exactly what nobody knows. One equation, two unknowns. Classic KMV closes the gap with a second equation linking equity and asset volatility; the deck notes those joint solutions run less stable · deck s93. This method iterates instead, which is the EM algorithm doing what it always does — alternate between recovering the hidden data and re-estimating the parameter:

The loop contracts fast — two to four rounds and σ_A stops moving. The recovered path sits above the equity path by the value of the debt claim, and the default point steps only on quarter ends, because that is when the filings change.

With the asset path in hand, two summary numbers fall out: the annualised drift R_A and the volatility σ_A. (The deck's literal mean is short by a factor of √250 — the engine annualises, which is the form that reproduces the professor's own workbook.) Distance to default asks the natural question: how many volatility-units of bad luck would it take to eat the whole cushion?

DD is honest about what it contains: the drift η_A sits in its numerator, and drift estimated from a stock-price window is the least stable number in the whole pipeline. The deck compresses the diagnosis to five words — η_A makes both unstable · deck s96 — both meaning DD and the probability read from it. Hold that thought for section 07.

Merton's closed form reads default off the endpoint: EDF = Φ(−DD) is the chance the assets sit below the debt at maturity. But a borrower who spends March through September under water and recovers by December has still defaulted in every sense a lender cares about. The first-passage view counts those paths: default is the first time the asset path touches the barrier, not where it happens to end.

This is the PIT PD the site's cards show. It is the more conservative of the two — and, for a bank that must mark a loan the day it sours, the more truthful one.

KMV worked. It read real credit information out of market prices — Moody's paid $220 million for it. The deck's verdict is double-edged: it is a Point-In-Time forward looking rating, and its most significant problem: unstable rating with huge volatility · deck s71. It moves whenever the stock moves, and a lender does not re-underwrite a loan because Tuesday was volatile. In Prof. Yang's telling, the complaint from every bank was the same: a warning that arrives days before the failure is no warning at all — and a rating that upgrades on Monday and downgrades on Thursday cannot be planned against.

The stock's behaviour and the credit's behaviour are different things. The task is to subtract the first and keep the second.

The paper's move is to describe a credit by its behaviour rather than by any one probability: μ = E[τ], how long the borrower is expected to live, and CCM — the Credit Corrosion Measure — how violently that lifetime is spread. The rating is their ratio:

What remains is to say it in S&P's language. The paper's rule is that a conversion must not manufacture regulatory arbitrage: the loss reserve (the PD) and the capital confidence level (α) must both survive the translation. α turns out to depend on CCM alone — so the recipe is: compute the first-passage confidence level, find the S&P corrosion CCM* that carries the same α, evaluate the S&P RiskScore there, and read Table 8.

Here is the whole chain run on KO's latest build — the same chain every company page prints, and every hop below opens its derivation:

The outlook is the last, quietly contrarian touch: Outlook = PD_FH − TTC PD. If point-in-time pressure sits above the cycle level and risk reverts to the cycle, the pressure is expected to decay — so PIT above TTC reads as positive · paper prop 5.3.

Large caps flatter. When market equity dwarfs the default point, DD runs to ten-plus volatilities, every PD underflows, and most of the mega-cap universe lands AAA. AT&T rates AAA here while the agencies hold it around BBB — the structural read sees an enormous equity cushion and cannot see covenant, sector, or management risk. That is a property of the Merton family, not a bug in this build; the ratings are most informative where the cushion is thin.

Default ratings only. The model prices the chance of failure, not the recovery after it — senior unsecured, no LGD · deck s45.

The teaching deck has known misprints. Two matter: the annualisation of R_A (short by √250 as literally printed) and one worked TiC value that cannot be reproduced from its own inputs. The engine follows the paper's forms and the professor's answer workbook, and the project's knowledge base documents every discrepancy with page numbers — a dashboard that hides its sources' rough edges would be advertising, not research.

The fine scale is a course artifact. Grades like “AAA-” do not exist on the real S&P scale; they come from the course's fine bucketing of TTC PD and are kept verbatim so the site never disagrees with the model output.

Data has seams. Deep history is Yahoo's chart API spliced onto Massive's window (they agree to 0.0001% on the overlap); filings arrive with a lag and step on quarter ends; the risk-free series is FRED's DGS1. The appendix on every company page traces each number to its source.