How does survival analysis model the timing of credit defaults, and what is the hazard rate?

Survival analysis models the time until an event — in credit risk, that event is default. Unlike logistic regression which predicts a binary outcome at a fixed horizon, survival analysis captures when default is most likely to occur.

Core Functions

1. Survival function S(t): The probability of surviving (not defaulting) beyond time t.
2. Hazard function h(t): The instantaneous default rate at time t, conditional on having survived to that point. Think of it as the conditional intensity of default.
3. Cumulative hazard H(t): The integral of h(t) from 0 to t.

The key relationship: S(t) = exp(-H(t))

If the hazard rate is constant (h(t) = lambda), then defaults follow an exponential distribution: S(t) = exp(-lambda * t). This is the simplest case and commonly tested on the FRM.

Example: Evergreen Credit Partners' Loan Portfolio

Evergreen Credit Partners estimates a constant annual hazard rate of lambda = 3.5% for its middle-market loan portfolio. What is the probability a loan survives 5 years without defaulting?

S(5) = exp(-0.035 x 5) = exp(-0.175) = 0.8395 or 83.95%

The 5-year cumulative default probability is: 1 - 0.8395 = 16.05%

The probability of defaulting specifically in year 3 (conditional on surviving to year 2):

P(default in year 3 | survive to year 2) = 1 - exp(-0.035 x 1) = 3.44%

Notice this equals 1 - exp(-lambda) regardless of when we start — that is the memoryless property of the exponential distribution.

Non-Constant Hazard Rates

In practice, hazard rates vary over time. For corporate bonds, the pattern often looks like:

• Low hazard initially (newly issued, strong cash position)
• Rising hazard at 2-4 years (debt burden, cyclical pressures)
• Declining hazard at 7+ years (survivorship bias — weak firms already defaulted)

This hump-shaped hazard is captured by the Weibull distribution or Cox proportional hazards model, both of which extend the constant-hazard framework.

Connection to Credit Spreads

Under risk-neutral pricing, the hazard rate is approximately:

> h(t) ≈ Credit Spread / (1 - Recovery Rate)

If a BBB bond trades at a 180 bps spread over Treasuries and the assumed recovery rate is 40%, the implied hazard rate is: 0.018 / 0.60 = 3.0% per year.

FRM exam tip: Know the constant hazard case cold — compute survival probabilities, cumulative default probabilities, and conditional default probabilities. Also understand that the Cox proportional hazards model allows covariates (like leverage, sector) to shift the baseline hazard multiplicatively.

For more on credit risk modeling, check our FRM Part I study materials.