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Distinguish default probabilities that are conditional from those that are unconditional, defining each.

FRM® Part I · Valuation and Risk Models · External and Internal Credit Ratings · FRM1.VR.4.b

The idea

A conditional default probability is the chance of defaulting in a given period, given that the issuer has survived to the start of that period. An unconditional default probability is the chance, assessed today, of defaulting in a specified future period, with no assumption that the issuer survives until then. The two are linked by a simple rule. Unconditional PD = conditional PD × probability of surviving to the start of the year. The unconditional probabilities across years add up to the cumulative default probability. Cumulative PD = 1 − the product of (1 − conditional PD) over the years. In year 1 the two coincide, because survival is certain. The usual mistake is to divide by the wrong survival probability, or to treat a conditional rate as if it were an unconditional one. Exam questions give one of these and ask for the other.

Formulas

Unconditional default probability for a year

unconditional PD = conditional PD × S

conditional PD
the probability of default in the year, given survival to its start
S
the probability of surviving to the start of the year

In year 1 the two coincide because S is 1. Dividing by the wrong survival probability is the usual slip when going the other way.

Cumulative default probability from conditional rates

cumulative PD = 1 − Π(1 − conditional PDₜ)

Π
the product over each year up to the horizon
conditional PDₜ
the conditional default probability in year t

Survival multiplies, so cumulative PD is one minus joint survival. Adding the conditional rates overstates it.

A practice question

From the question bank, with the reason each wrong option is wrong.

A conditional default probability for year 3 is best defined as the probability that the issuer:

  1. defaults during year 3, as assessed today, without conditioning on survival to year 3.Why not: This is the unconditional probability; the conditional version restricts to issuers that survive to year 3.
  2. defaults at any time up to the end of year 3.Why not: This is the cumulative default probability, which covers all years up to year 3.
  3. defaults during year 3, given that it has survived to the start of year 3. - correct
  4. survives through the end of year 3, given that it started year 1 solvent.Why not: This is the survival probability, the complement of the cumulative default probability, not a conditional default probability.

Conditional default probability is the default probability in a period given survival to the period's start. It is a probability among survivors, while the unconditional probability is measured from today.

A harder one

An issuer has a conditional default probability of 5% in year 1 and 8% in year 2. The unconditional probability of default in year 2 is closest to:

  1. 7.6% - correct
  2. 8.0%Why not: Reports the conditional probability; the unconditional figure must be multiplied by the 95% survival probability to year 2.
  3. 12.6%Why not: Reports the two-year cumulative default probability, 1 - 0.95 x 0.92, not the year-2 probability alone.
  4. 0.4%Why not: Multiplies the two conditional probabilities (5% x 8%) as if default had to happen in both years.

Unconditional PD(year 2) = survival to year 2 x conditional PD = (1 - 0.05) x 0.08 = 0.076, or 7.6%.

The hardest tier

A bond's cumulative default probabilities are 12% by the end of year 3 and 18% by the end of year 4. The conditional probability of default in year 4, given survival through year 3, is closest to:

  1. 6.00%Why not: Takes the increase in cumulative probability, which is the unconditional year-4 probability, without dividing by survival to year 3.
  2. 7.32%Why not: Divides the increase by survival to the end of year 4 (82%) instead of survival to the end of year 3 (88%).
  3. 6.82% - correct
  4. 4.50%Why not: Divides the four-year cumulative probability by four, spreading it evenly across years.

Unconditional PD in year 4 = 18% - 12% = 6%. Survival to the start of year 4 = 88%. Conditional = 6 / 88 = 6.82%.

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