Default probability by the end of the second year, denoted as $Q(2)$, is calculated using the unconditional default probability formula

$$Q(t) = 1 - e^{-ht}$$

, where $h$ is the hazard rate and $t$ is the time in years.

Plugging in the numbers:

$$Q(2) = 1 - e^{-0.02 \times 2} = 1 - e^{-0.04} \approx 0.0392$$

B is incorrect. It reflects the default probability by the end of the first year, not the second year.
C is incorrect. This is the complement of the default probability, representing the survival probability to the end of the second year.
D is incorrect. This is the complement of the default probability for the first year.

Things to Remember.

• Default probabilities are derived from exponential functions of the hazard rates over time.
• Hazard rates implied from market spreads provide a quantifiable annual risk of default for the valuation of CDS.
• Understanding the calculation of default probabilities is vital for determining both the CDS price and expected payments.