Cumulative Default Rates and Mortality Tables

Life insurers have used mortality tables for centuries to track what fraction of a group of people are still alive at each age. Credit analysts borrowed the exact same idea for bonds: take a group of bonds issued in the same year at the same rating, and track what fraction of the original pool has defaulted by each year since issuance, rather than trying to guess a single "default rate" that applies to every year equally.

Building the table

Each year's marginal default rate is calculated only against bonds that survived to the start of that year, not against the original pool size — a bond that already defaulted in year 2 can't default again in year 3. Cumulative survival compounds the marginal survival rates (one minus the marginal default rate) across each year:

$$S_n = \prod_{i=1}^{n} (1 - d_i)$$

In words: multiply together each year's survival rate (one minus that year's marginal default rate) to get the fraction of the original pool still performing after $n$ years. The cumulative default rate is simply $1 - S_n$.

Worked example

A cohort of BB-rated bonds issued in the same year has these marginal default rates: year 1, 2%; year 2, 3%; year 3, 2.5%.

1. Year 1 survival. $1 - 0.02 = 0.98$.
2. Year 2 survival (compounded). $0.98 \times (1 - 0.03) = 0.98 \times 0.97 = 0.9506$.
3. Year 3 survival (compounded). $0.9506 \times (1 - 0.025) = 0.9506 \times 0.975 \approx 0.9268$.
4. Cumulative default rate by year 3. $1 - 0.9268 = 7.32\%$.

Note this is less than simply adding the marginal rates ($2\% + 3\% + 2.5\% = 7.5\%$), because each year's default rate applies only to the shrinking pool of survivors, not the original full cohort.

What this means in practice

Rating agencies publish mortality tables by cohort and rating to give investors an empirical, age-specific default curve rather than a single average number — useful for pricing a bond's expected loss at a specific point in its life, not just over its full lifetime.