Options, Futures & Other Derivatives, - John C. Hull

From Ratings to Historical Default Probabilities

May 2, 2026neoinsight4 min read

A credit rating — AAA, BBB, and so on — is a compact measure of relative credit risk. It shows how risky one debtor is compared with another. Therefore, for the rating to become analytically useful, it has to be connected to historical default data.

Cumulative Default Rate

A historical default table shows how frequently bonds with a given initial rating defaulted over different time horizons.

In other words, if a bond has a certain rating today, the table may show what percentage of similar bonds defaulted by the end of the first year, third year, fifth year, tenth year, and so on. This is the cumulative default rate.

It answers the question:

Of the bonds that started the period with this rating, what percentage had already defaulted by a specific point in time?

Two Interesting Observations from the Table

1. Default risk does not accumulate at the same speed across all ratings

The cumulative default rate always increases over time because defaults accumulate. But the speed of increase differs by rating.

For stronger issuers, default risk usually accumulates gradually. They are financially healthy at the starting point, so their short-term default risk is low.

For weaker issuers, risk is often concentrated in the early years. The first few years may be decisive. If the company survives this period, the remaining survivor group may be stronger than the original group.

So the table does not only show that “higher rating = lower risk.” It also shows the shape of credit risk through time.

2. The investment-grade boundary matters

The move from investment grade to speculative grade is not merely a symbolic downgrade.

Historically, the jump in default rates around the BBB/BB boundary is quite large. This is why losing investment-grade status can have a serious impact on a company’s cost of funding, investor base, and refinancing flexibility.

The table shows that credit risk does not always rise smoothly, one rating notch at a time. At some boundaries, the change is economically meaningful.

Annual Unconditional Default Probability

The cumulative default rate shows how many bonds had defaulted by a specific point in time.

But we may want to ask a different question:

What percentage defaulted specifically during this year?

This is the annual unconditional default probability.

It is calculated as the difference between two cumulative default rates:

$q_t = Q_t – Q_{t-1}$ Where:

$Q_t$ Qt = cumulative default probability by the end of year $t$ t
$Q_{t-1}$ Qt−1 = cumulative default probability by the end of the previous year
$q_t$ qt = probability of default during year $t$ t, viewed from today

For example, if the cumulative default probability is 29.384% by the end of year 2 and 38.682% by the end of year 3, then the probability of default during year 3 is:

$38.682\% – 29.384\% = 9.298\%$ This means:

From today’s perspective, 9.298% of the original group will default specifically during the third year.

Conditional Default Probability — Hazard Rate

The annual unconditional default probability is measured from the original starting point. But suppose that by the start of year 3, some bonds have already defaulted. In that case, a more precise question appears:

If the bond has survived until the beginning of year 3, what is the probability that it defaults during year 3?

This is the annual conditional default probability, or the discrete version of the hazard rate.

$h_t = \frac{Q_t – Q_{t-1}}{1 – Q_{t-1}}$ Using the same example: $h_3 = \frac{9.298\%}{70.616\%} = 13.17\%$

So the same default event can be viewed in two ways:

Measure	Meaning
9.298%	Probability of default during year 3, viewed from today
13.17%	Probability of default during year 3, conditional on the bond having survived until year 3

This is one of the most useful ideas in credit analysis. It transforms a cumulative default table into a year-by-year picture of conditional credit risk.

Translating Annual Hazard Rates into Shorter Periods

Once we have an annual conditional default probability, we can translate it into shorter periods — for example, quarters or months. This is useful because bonds and other credit instruments often have interim cash flows before year-end.

For example, if the annual conditional default probability is 12%, then the monthly probability can be calculated as: $p_{month}=1-(1-12\%)^{1/12}$

This requires one simplifying assumption:

Default risk is distributed evenly within the year.

This assumption may not be perfect, but it is practical. It allows us to estimate monthly or quarterly credit risk from annual rating data without needing separate monthly historical default tables.

From Probability to Expected Loss

Default probability is not the same as expected loss.

If a bond defaults, the investor may still recover part of the face value. The recovered percentage is the recovery rate.

Loss severity is: $LGD = 1 – Recovery\ Rate$ Expected loss is: $Expected\ Loss = PD \times LGD$

Therefore, a 3% default probability and a 40% recovery rate imply a 1.8% expected loss:

$3\% \times 60\% = 1.8\%$

The recovery rate also depends on the instrument’s position in the capital structure. Senior secured debt, senior unsecured debt, subordinated debt, and junior claims can have very different recovery outcomes.

Excel-credit_risk_default_tables_model

Adapted from:
Options, Futures, and Other Derivatives — John C. Hull