Default probability (PD) is the probability that a borrower or counterparty experiences a specified default event within a stated horizon. A PD helps turn a possible credit loss into a probability-weighted exposure to that loss. It says how likely default is; loss given default and exposure at default describe how much is lost if it happens.
“PD is 2%” leaves several questions unanswered: by when, under what default definition, conditional on what information, and under which probability measure? Those details determine whether the number belongs in a loss forecast, a pricing model or neither. A downgrade in credit rating is not automatically a default.
A one-year horizon is common, rather than built into the mathematics. For example, the Austrian FMA's CRR glossary definition specifies a one-year period. A multi-year bond or loan also needs the probability of surviving, or first defaulting, over longer horizons.
Three probabilities behind “year two”
Consider a fictional borrower that is performing today. Stipulate a 2% conditional default probability in each year, given survival to the start of that year. This is a fixed illustrative model, with no claim about a real issuer or a rating category. Default means first occurrence of the stipulated event; once it occurs, the borrower leaves the surviving population.
| Quantity | Denominator and event | Probability |
|---|---|---|
| Conditional year-two PD | Default during year two, among those surviving year one | 2% |
| Unconditional year-two increment | First default during year two, measured from today's population | |
| Cumulative two-year PD | Default in either year, measured from today's population | |
| Two-year survival | No default in either year |
The conditional probability stays at 2%; the second year's unconditional increment is smaller because only 98% of the original population reaches that year without default. Adding 2% and 2% uses different denominators. The answer 4% is an approximation: it exceeds the exact cumulative PD by 0.04 percentage points.
Exact model weights per 10,000 initial borrowers under a stipulated 2% annual conditional PD. The weights explain denominators; they are neither realised default counts nor a simulation.
Multiplying the survival probabilities uses the probability chain rule. It does not require a borrower to default independently in successive years: first default can occur only once. What is assumed here is that the conditional annual PD remains unchanged after survival. In practice, changes in credit quality or the economic environment can change it.
From annual PDs to a default curve
Let be the first default time, the survival function, and the cumulative PD, all viewed from today for a borrower not yet in default. For integer year , define
Then
The unconditional probability of first default during year is
This increment, rather than the conditional alone, is the probability weight for that year's first-default loss when valuing or forecasting from today. Exposure, recovery, discounting and any dependence between them still need their own assumptions. A cumulative PD is not a loss percentage.
A hazard rate expresses conditional default risk per unit time. For a constant hazard in a continuous-time model,
With time in years, an annual PD of 2% corresponds to per year, or about 2.0203% per year. A hazard of 2% per year instead gives an annual PD of . The difference is small here, but intensity and finite-horizon probability are different quantities.
For the continuous-time derivation and the move into bond and CDS valuation, read Reduced-Form Models: Hazard Rates and Risk-Neutral Default Probabilities. The linked page is a public preview; the full lesson requires membership.
A forecast PD and a pricing PD answer different questions
A physical PD, under , estimates default frequency in the real world using historical experience, borrower information and a forecasting model. The estimate depends on the population and economic conditions represented; a past default rate is evidence for an estimate, not certainty about the next period.
A risk-neutral default probability, under , is a pricing-model quantity. It is calibrated to prices subject to assumptions about recovery, discounting and the instrument's cash flows. A credit spread also reflects compensation for risk and may contain liquidity and other effects, so reading a spread as a direct physical default forecast can mislead. There is no universal conversion factor between the two measures.
The Federal Reserve Bank of Chicago's CDS analysis explicitly identifies its price-implied probability as risk-adjusted and explains why risk aversion affects it. Keep the measure, horizon and default event attached to every quoted PD. The survival arithmetic above works under either measure, provided all its inputs use the same one.
Practice: the second year becomes riskier
A fictional performing borrower has a 5% first-year PD. Conditional on surviving that year, its second-year PD is 10%. What are the unconditional probability of first default during year two and the cumulative two-year PD? Use this stipulated model without discounting or a recovery calculation.
Answer. Year-one survival is 95%, so the second-year increment is . Cumulative PD is , equivalently . Adding 5% and 10% would ignore that the second number applies only to survivors. Among those survivors, the second-year PD remains 10%.
Sources
- John C. Hull, Options, Futures, and Other Derivatives, ninth Global Edition, Pearson, 2018, §24.2, pp. 567–568 (unconditional versus conditional default, survival and hazard), and §24.5, pp. 572–575 (historical versus risk-neutral estimates and their uses).
- Austrian Financial Market Authority: Probability of Default (PD), the one-year CRR definition; checked 27 October 2026.
- Federal Reserve Bank of Chicago: What Does the CDS Market Imply for a U.S. Default?, sections “What do CDS premiums tell us about the likelihood of a default?” and the concluding risk-adjustment caveats; checked 27 October 2026.