Cumulative, marginal, conditional, annualised: demystifying the PD term structure

A five-year surety bond, a three-year credit policy, a lifetime IFRS 9 provision. Each needs a probability of default over more than one year, and each is regularly mispriced because four different numbers all get called “the PD”.

Start with the default time

Let $\tau$ be the time at which an obligor defaults. Everything else follows from two functions of it: the cumulative PD and its complement, the survival probability.

The four PDs

Cumulative PD, $\text{CPD}(t)$: the probability of defaulting at any point up to year $t$. This is what rating agency default studies report.

Marginal (unconditional) PD for year $t$: the probability, seen from today, of defaulting in year $t$ specifically.

Conditional (forward) PD for year $t$: the probability of defaulting in year $t$, given the obligor survived to the start of it. This is the one-year PD a surviving obligor carries each year.

Survival chains through the conditional PDs:

Annualised PD over a horizon $T$: the constant yearly PD that would produce the same cumulative PD. It is a quoting convenience for comparing tenors, not the probability of anything that happens in a particular year.

In continuous time the same idea is a hazard rate. With a constant hazard $\lambda$:

One obligor, all four numbers

Take a sub-investment-grade, mid-market obligor whose conditional PD drifts up from 2.0% to 2.6% over five years (illustrative figures):

Read across year 5. The conditional PD is 2.60%, but the marginal PD is lower, 2.36%, because only 91% of obligors are still around to default. Marginal PDs can fall while conditional PDs rise. The cumulative PD is 11.44%, not the 12.00% you get by adding up the yearly numbers. And the annualised figure, 2.40%, is a blend that never occurs in any single year.

Which one to use, where

Pricing and lifetime expected loss use marginal PDs. For a multi-year credit or surety exposure:

with the exposure profile $\text{EAD}_t$ (a surety bond’s exposure often falls as the contract completes) and a discount factor $\text{DF}_t$. IFRS 9 lifetime expected credit loss has exactly this structure.

Monitoring a live book uses conditional PDs. Next year’s risk for obligors still on the books is $\text{cPD}$, not $\text{mPD}$.

Comparing tenors uses annualised PDs or hazard rates. Benchmarking against agency studies uses cumulative PDs.

The usual mistakes

Multiplying the one-year PD by the tenor. Harmless for tiny PDs, increasingly wrong as PDs and tenors grow, because it ignores that an obligor can only default once.

Treating conditional PDs as marginal, which overstates expected loss in the later years.

Assuming a flat term structure. For strong credits the conditional PD typically rises with tenor: there is more time for things to go wrong. For weak credits it often flattens or falls, because the survivors are the ones that improved. Rating migration matrices capture this; a flat curve does not.

Mixing philosophies carelessly. A point in time PD for the first year that converges towards the long-run curve as the forecast horizon fades is sound. Using a point in time PD for every year of a ten-year bond assumes today’s economy lasts a decade. I unpack that distinction in Through the cycle vs point in time PD.

Banu Munnangi is SVP, Credit Reinsurance AI at Marsh Re, where he leads data and AI for the Credit, Bond and Political Risk specialty. More writing · Expertise

Views are my own and do not represent my employer. Examples use illustrative figures, not client or employer data.