The table below shows the historical cumulative probabilities for corporate bonds with quality rating of AA:

Cumulative Default Probabilities (%)
Year 1 2 3 4 5
AA 0.10 0.40 1.10 1.50 1.76

Calculate the marginal (conditional) probability of default in the 5th year (from t=4 to t=5).

(Round to 4 decimal places.)

Question

The table below shows the historical cumulative probabilities for corporate bonds with quality rating of AA:

Cumulative Default Probabilities (%)
Year	1	2	3	4	5
AA	0.10	0.40	1.10	1.50	1.76

Calculate the marginal (conditional) probability of default in the 5th year (from t=4 to t=5).

(Round to 4 decimal places.)

BlackTom AI · Accepted Answer

We start by identifying what the question calls a marginal (conditional) probability of default in year 5: it is the probability that default occurs during the 5th year, given the information up to the end of year 4. The cumulative probabilities are given as percentages: P(default by year 4) = 1.50% and P(default by year 5) = 1.76%. The marginal probability of default in year 5 is found by subtracting the cumulative probability at t=4 from the cumulative probability at t=5:

P(default in year 5) = P(default by year 5) − P(default by year 4) = 1.76% − 1.50% = 0.26%.

Converting to a decimal for the answer, 0.26% = 0.0026. Rounding to 4 decimal places yields 0.0026.

Interpreting the result: this value represents the unconditional probability that a bond with AA rating defaults specifically during the 5th year (from t = 4 to t = 5), assuming the data are cumulative defaults up to each year. If one wanted a conditional probability given survival to the end of year 4, one would divide this year-5 default probability by the survival probability to year 4, but the question asks for the marginal (unconditional) probability for that year.

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