How do I use the hazard rate to calculate cumulative default probability over multiple years?

The hazard rate framework is elegant and deeply testable at Level II. The key is understanding the relationship between conditional and cumulative probabilities.

Definitions:

• Hazard rate (h): The probability of default in a given year, conditional on having survived to the beginning of that year
• Survival probability: Probability of NOT defaulting through a given period
• Cumulative default probability (PD): Probability of defaulting at any point during the entire period

The Key Relationships:

Survival probability through year t:

S(t) = (1 - h)^t (if hazard rate is constant)

Cumulative default probability:

PD(t) = 1 - S(t) = 1 - (1 - h)^t

Why It's Not Simply h x t:

The hazard rate compounds — each year's default probability applies only to the survivors from the previous year. This is identical to compound interest logic.

Worked Example:

Constant hazard rate h = 3% per year

Calculation Check:

• S(5) = (1 - 0.03)^5 = (0.97)^5 = 0.8587
• PD(5) = 1 - 0.8587 = 14.13% (not 15%!)

The cumulative PD (14.13%) is less than the naive calculation (5 x 3% = 15%) because each year's default probability applies to a shrinking population of survivors.

Variable Hazard Rates:

When the hazard rate changes over time:

S(t) = (1-h_1)(1-h_2)...(1-h_t)

Example: h_1 = 2%, h_2 = 3%, h_3 = 5%

S(3) = (0.98)(0.97)(0.95) = 0.9028

PD(3) = 1 - 0.9028 = 9.72%

Marginal (Unconditional) Default Probability:

The probability of defaulting in year t specifically (not conditional on surviving):

Marginal PD(t) = S(t-1) x h_t

Year 3 marginal PD = 0.98 x 0.97 x 0.05 = 4.75% (not 5%, because some entities already defaulted)

Exam Application — Bond Pricing:

The hazard rate framework connects directly to credit spreads:

Credit spread ≈ h x LGD = h x (1 - Recovery Rate)

If h = 3% and recovery = 40%:

Credit spread ≈ 3% x 60% = 1.80% = 180 bps

This provides a quick sanity check: if a BB bond trades at 250 bps but the hazard rate implies only 180 bps of credit spread, the extra 70 bps reflects liquidity premium.

Exam Tip: Practice computing cumulative PD, survival probabilities, and marginal PD from both constant and variable hazard rates. The connection to credit spreads (spread ≈ h x LGD) is frequently tested.

Master credit modeling in our CFA Level II fixed income modules.