Survival Analysis & Time-to-Default
Binary default models answer whether a borrower defaults. Survival analysis adds the missing dimension: when. That time structure is central for lifetime PD, IFRS 9 staging logic, and any model that needs a full default term structure instead of a single 12-month probability.
Three core functions
Survival function S(t)
The probability that the borrower survives beyond time t without defaulting.
It starts at 1 and declines over time as default risk accumulates.
Hazard function h(t)
The instantaneous default intensity at time t, conditional on survival up to t.
This is the local speed of default risk.
Cumulative hazard H(t)
The accumulation of hazard over time.
It links directly to survival and therefore to cumulative PD.
Why censoring changes everything
The core problem
In credit data, many borrowers do not default during the observation window. You know they survived at least until a certain date, but not what happens afterwards.
That is right-censoring. Ignoring it means losing valid time information and biasing the survival estimate.
Why logistic regression is not enough
A simple default / non-default model treats a 1-year non-default and a 10-year non-default identically.
Survival analysis uses the time dimension properly: how long the borrower stayed alive in the portfolio before default or censoring.
A useful order for learning survival analysis
Start with survival and hazard intuition
Before formulas, understand that one tells you how much survives and the other tells you how fast risk is arriving.
Then understand censoring
Most of the value of survival models appears precisely because some outcomes are incomplete at the observation date.
Then move to non-parametric structure
Kaplan-Meier gives the empirical survival curve before you start imposing functional assumptions.
Then add covariates and IFRS 9 logic
Once the time structure is clear, Cox PH and lifetime PD term structures become much easier to interpret.
Kaplan-Meier survival curve
Kaplan-Meier is the cleanest non-parametric estimator of S(t). Event times create downward steps. Censored observations stay in the risk set until they leave, but do not generate a drop themselves.
Hazard shape determines the PD term structure
Different portfolios do not share the same time profile of risk. Some have nearly constant hazard, some season downward, some rise with age, and some show a hump or bathtub structure.
Cox PH — adding borrower covariates
Cox PH allows covariates to shift the hazard up or down without forcing a specific baseline hazard shape. That makes it one of the most practical bridges between classical survival analysis and modern lifetime PD modelling.
From survival to lifetime ECL term structures
This is the practical endpoint for many credit-risk use cases. The survival curve turns into cumulative PD, marginal PD, and forward PD — exactly the quantities that drive lifetime ECL and stage-dependent measurement.
PD term structure — cumulative, marginal, forward
Parameters
PD term table
Survival models compared
| Model | Type | Hazard shape | Handles censoring? | Typical use |
|---|---|---|---|---|
| Kaplan-Meier | Non-parametric | Empirical | Yes | Exploratory survival estimation |
| Nelson-Aalen | Non-parametric | Empirical | Yes | Cumulative hazard estimation |
| Exponential | Parametric | Constant | Yes | Simple lifetime PD structures |
| Weibull | Parametric | Monotone ↑ / ↓ | Yes | Ageing or seasoning effects |
| Cox PH | Semi-parametric | Flexible baseline | Yes | Lifetime PD with covariates |
| AFT / Log-logistic | Parametric | More flexible | Yes | When PH is doubtful |
| Discrete-time hazard | Semi-parametric | Period-based | Yes | Monthly / quarterly default modelling |
Concepts every validator should keep
Proportional hazards may fail
If macro or behavioural effects change over time, a simple Cox PH assumption can be too rigid.
Default is not the only exit
Prepayment, maturity, cure, and restructuring all compete with default and can distort naive lifetime estimates if ignored.
Term structures depend on philosophy
TTC curves reflect long-run average behaviour. PIT curves absorb current and forward-looking macro conditions.
AUC has a survival analogue
The concordance index plays the role of discrimination summary for censored time-to-event models.
Calibration is not just 12-month
A survival-based lifetime PD model should be checked across multiple horizons, not only at the first year.
Cure matters too
In some portfolios, default is not an absorbing endpoint. Cure and re-default can materially change the expected loss path.
What to leave this page with
Survival analysis adds the missing question to default modelling: not only whether default happens, but when.
The useful order is: first understand survival, hazard, and censoring; then read the empirical Kaplan-Meier curve; then interpret hazard shapes; then add covariates through Cox PH; then translate the result into IFRS 9 lifetime PD structures.
Once that structure is clear, lifetime PD stops looking like an abstract regulatory requirement and starts looking like a natural time-to-event model.