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 1.71

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	1.71

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

First, identify what the cumulative default probabilities represent: the probability of default by the end of each year for AA-rated bonds. The given table shows cumulative defaults at year 4 and year 5 as 1.50% and 1.71%, respectively.
Next, recall that the marginal (yearly, or conditional) default probability for year t equals the increase in the cumulative default probability from year t-1 to year t. In mathematical terms: P(default in year 5) = P(default by end of year 5) − P(default by end of year 4).
Plug in the values from the table: 1.71% − 1.50% = 0.21%.
Convert the result to decimal form as required by the instruction: 0.21% = 0.0021 in decimal.
Since the instruction asks for rounding to 4 decimal places, 0.0021 already has four decimal places (0.0021).
Thus, the marginal probability of default in the 5th year (from t=4 to t=5) is 0.0021 when expressed in decimal form, which corresponds to 0.21% in percentage terms.

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