Every derivative you've priced so far in this course — a bond option, a swap, a cap — has quietly assumed that whoever owes you money will actually pay it. A credit trading desk cannot afford that assumption. Its entire job is to put a number on the chance that a borrower — a corporation, a bank, a sovereign — fails to pay what it owes, and then to price that number into everything: the yield demanded on a bond, the premium on a CDS, the capital a bank must hold against its loan book. This lesson builds the first and most basic version of that number: a default probability, read first off a rating agency's own scorecard and then rebuilt from a single clean assumption about how default arrives.

The rating scale

Three agencies — S&P, Moody's, and Fitch — dominate corporate and sovereign credit ratings. Each assigns a letter grade meant to summarize, in one symbol, an issuer's ability and willingness to pay its debts in full and on time. The scales differ slightly in notation but agree on the ordering:

The line between BBB− (or Baa3) and BB+ (Ba1) is the single most economically important boundary on the whole scale: it separates investment grade from speculative grade. Many pension funds, insurers, and money-market funds are contractually barred from holding anything below it — so a single-notch downgrade across that line can force forced selling worth billions of dollars, independent of anything else changing about the issuer.

From letters to numbers: historical default rates

A rating is an ordinal ranking — AAA is safer than AA, which is safer than A — but a trading desk needs a number. Rating agencies publish exactly that, tracking every bond they've ever rated and reporting, for each rating and time horizon, what fraction actually defaulted. The table below shows illustrative long-run figures of the kind these studies produce (cumulative probability of default, in percent, from the start of the horizon):

Two patterns matter more than any single cell. First, reading down a column, default probability rises steeply as rating falls — a CCC-rated borrower is roughly a thousand times more likely to default within a year than an AAA one. Second, reading across a row, default probability rises with horizon for every rating: the longer you're exposed, the more chances the world has to go wrong. Both patterns are exactly what a pricing model needs to reproduce.

The hazard-rate model: default as a constant "chance per instant"

A historical table is a lookup, not a model — it can't tell you the 7-year default probability if you only have 1, 5 and 10-year figures. The standard trick is to model default the same way physicists model radioactive decay: assume a constant instantaneous hazard rate \lambda (also called the default intensity) — the conditional probability of default in the next tiny instant, given survival so far, per unit time. With \lambda constant, the probability of surviving from now to time t follows an exponential decay:

A bigger \lambda means faster decay of the survival curve — a riskier borrower. The chart below plots S(t) for a hazard rate you control; drag the slider and watch how much a "small-looking" change in \lambda compounds over a long horizon.

Worked example: calibrating \lambda to a rating

Take the BBB row above: a 5-year cumulative default probability of 1.8\%. Solving 1 - e^{-5\lambda} = 0.018 for \lambda:

Now use that calibrated \lambda to predict the 10-year cumulative default probability, and compare it with the table's actual entry:

The table says 4.5\% — noticeably higher than the 3.57\% a constant hazard rate predicts. That gap isn't a mistake in the arithmetic; it's a real limitation of the constant-\lambda assumption, which the "Watch out!" box below unpacks.