B is correct. To calculate conditional PD in year 3, one must first calculate the unconditional PD in year 3. This equals the difference between the cumulative PDs in year 3 and year 2:

$39.86% - 34.51% = 5.35%$

Conditional PD in year 3 is the PD in year 3 on the condition that the bond survives to year 3, or the unconditional PD divided by the survival rate. The survival rate to year 3 is one minus the cumulative PD to year 2: $100% - 34.51% = 65.49%$. Conditional PD in year 3 is therefore:

$5.35% / 65.49% = 8.17%$

• A is incorrect. This is the year 3 unconditional PD.
• C is incorrect. This plugs the year 2 and year 3 cumulative PDs into the formula for calculating the year 3 unconditional PD from hazard rates: $exp(-\bar{h}_1 t_1) - exp(-\bar{h}_2 t_2) = exp(-34.51\% \cdot 2) - exp(-39.86\% \cdot 3) = 19.90\%$
• D is incorrect. This multiplies the survival rate to year 3 by the year 3 cumulative probability.

Learning Objective: Define conditional and unconditional default probabilities and explain the difference.