Outline how a borrower's rating relates to its probability of default.
The idea
A rating is an ordinal ranking of relative default risk. It becomes a probability of default only through calibration to history, since agencies publish the fraction of issuers in each rating that defaulted within one, two, five or more years. Default probabilities rise as the rating worsens, slowly at the top of the scale and steeply in the speculative grades. The term structure also depends on the rating. For investment-grade issuers the conditional probability of default in a later year, given survival to it, tends to rise with the horizon, because today's strength may fade. For poorly rated issuers it tends to fall, because survivors of the early years are the stronger names. Let F(t) be the cumulative default probability by year t, and q the conditional probability in year t. The conditional probability is then given by the following. q = (F(t) − F(t−1)) ÷ (1 − F(t−1)). Extrapolating a one-year PD with a constant hazard rate misstates both ends of the scale.
Formulas
Conditional default probability in year t
conditional PD = (C(t) − C(t−1)) ÷ (1 − C(t−1))
- C(t)
- the cumulative default probability to the end of year t
- C(t−1)
- the cumulative default probability to the end of the previous year
Divides by survival to the start of the year. It tends to rise with the horizon for investment grade and fall for weak ratings.
A practice question
Moving from AAA down to CCC, how do average historical default probabilities for agency-rated issuers behave?
- They rise by roughly equal increments with each notch down the scale.Why not: Default probabilities are convex in the rating, not linear.
- They are essentially zero for every investment-grade rating and then jump to 100 percent at the speculative boundary.Why not: Investment-grade default rates are low but not zero, and the increase is gradual, not a jump to certainty.
- They rise, slowly in the investment-grade categories and steeply in the speculative grades. - correct
- They fall, because lower-rated issuers offer higher coupons that compensate for default.Why not: Coupons compensate investors but do not reduce default probabilities.
Default frequency rises roughly exponentially as the rating worsens, so each step down is a larger proportional increase.
A harder one
Historical cumulative default probabilities are: BBB 0.20% after one year and 0.55% after two years; CCC 25.0% after one year and 36.0% after two years. What are the year-two conditional default probabilities (given survival to the start of year two), and how do they compare with year one?
- BBB 0.35%, above its year-one value; CCC 14.67%, below its year-one value. - correct
- BBB 0.35%; CCC 11.00%, both equal to the year-two increments in cumulative probability.Why not: Uses the unconditional increment for CCC without dividing by the survival probability.
- BBB 0.55%; CCC 36.0%, the year-two cumulative probabilities.Why not: Reports cumulative probabilities as if they were year-two conditional probabilities.
- BBB 0.20%; CCC 25.0%, the same as year one.Why not: Assumes a constant hazard and ignores the term structure of default probabilities.
Conditional PD in year 2 = (cum2 - cum1) / (1 - cum1): BBB = 0.35%/0.9980 = 0.35%; CCC = 11.0%/0.75 = 14.67%. Investment grade rises with time, CCC falls.
The hardest tier
A CCC-rated issuer has a one-year default probability of 20%. Historical cumulative default probabilities for CCC are 20% (one year), 30% (two), 35% (three) and 38% (four). A risk manager extrapolates the 20% with a constant hazard rate to four years. What does this give, and what is the problem?
- 80.0%, which overstates the historical 38%, because the four annual probabilities were added.Why not: Adds the four annual probabilities instead of compounding survival; that gives 80%, not the constant-hazard figure.
- 59.0%, which understates the historical 38%, because conditional probabilities rise with the horizon.Why not: Right extrapolation but wrong conclusion: 59.0% is above 38.0%, and CCC conditional probabilities fall.
- 59.0%, which overstates the historical 38%, because conditional default probabilities for CCC issuers fall with the horizon. - correct
- 41.0%, which is close to the historical 38%, so the extrapolation is reliable for CCC.Why not: Reports the four-year survival probability rather than the cumulative default probability.
Constant hazard gives 1 - 0.80^4 = 59.0% against 38.0% observed. Year-4 conditional PD is (38-35)/65 = 4.6%, far below the 20% in year one: survivors are stronger.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Estimating Default Probabilities
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- Assess the quality of default probabilities that come out of the Kamakura model, the Merton model and the Moody's KMV model.