# Probabilities of Default

Key words: CDS spreads, Cumulative, Conditional (probability intensities) and Unconditional probability of default, Survival probability

Probability of default is a financial term describing the likelihood of a default over a particular time horizon. Measurement of the probability of default for a credit exposure over a given horizon is often the first step in credit risk modeling and pricing. For example, CVA is the process through which counterparty credit is priced and hedged. For CVA pricing, it is necessary to divide the remaining life of a derivative instrument into a number of time buckets and to calculate the unconditional probability of a counterparty default for each time bucket.
Term structure of credit spreads for the counterparty can be observed in the market. Such spreads can be used to estimate unconditional probabilities for a specific time bucket.

Widely used probabilities of default are:
Cumulative probability
Unconditional probabiblity
Conditional probabiltiy

The cumulative probabilities show the default chance through time.

Suppose that CDSs on SuperBank were trading at 50, 70, 80 and 100 basis points for 6, 12, 18 and 24 months respectively.
Using these quotes, cumulative PDs can be approximated as follows:

Cumulative PD = 1 – exp(-CDS quote * Maturity / LGD)

For the first bucket, the cumulative PD = 1 – exp(-0.5% * 0.5 / 60%) = 0.42%.

For example, the corporate bonds have 0.42% chance of defaulting after 6 months, 1.16% chance after one year, 1,98% chance after 18 months and 3,28% chance after 2 years.

The unconditional probabilities are the probabilities of default in a given bucket as viewed from time zero. The unconditional probability of a bond defaulting during year t is equal to the difference in the cumulative probability in one bucket minus the cumulative probability of default in another bucket tâˆ’1.

From the piicture above, the unconditional probabiltity of default in the second bucket is 1.16% – 0.42% = 0.74%.

The conditional probability is the probability of default in a given bucket conditional on no prior defaults.
This probability is equal to the unconditional probability of default in time t divided by the probability of survival at the beginning of the period.

The probability of survival is 100 minus the cumulative probability. For example, the probability that a
SuperBank will survive until the end of the third bucket is 98.02% (100 minus its cumulative probability 1.98%).

The probability that SuperBank will default during the time of the forth bucket conditional on no prior defaults is 1.32% (1.3% / 98.02%).