Survival Models in Credit: Predicting When Loans Default
Survival Models in Credit: Predicting When Loans Default
Two loan segments have the same 6% default rate over 24 months. In one, borrowers default around month 4; in the other, around month 20. The first loses $252,921 per 1,000 loans, the second $64,603. A binary scorecard rates them the same, and it can use only a third of a growing lender's loans to learn even that. Survival models predict when default happens, use every loan on the book, and supply what lifetime loss estimates require.
What you'll learn
Not if, but when
Most credit scorecards answer a yes-or-no question: will this borrower default within a fixed window, often 12 or 24 months? That framing, convenient for logistic regression and classification models, throws away two things lenders care about. It ignores when default happens, which decides how much the loan earned before it failed, and it ignores every loan that has not yet been on the books for the full window. Survival analysis, developed in medicine and engineering to model time until an event, answers both. Banasik, Crook and Thomas put the case for credit in a phrase: not if, but when will borrowers default.
Same default rate, different losses
Take two segments of $5,000, 36-month loans at 24% APR, monthly payment $196.16. Each has the same 24-month default rate, 6%. In one, defaults cluster around month 4; in the other, around month 20.
| Default timing | Payments received before default | Balance outstanding at default | Net loss per defaulted loan | Net loss per 1,000 loans |
|---|---|---|---|---|
| Month 4 | $785 | $4,604 | $4,215 | $252,921 |
| Month 20 | $3,923 | $2,663 | $1,077 | $64,603 |
A binary target cannot see the difference, and pricing built on it cannot either. Per our price decomposition analysis, the expected-loss term in a loan's price depends on exposure at default, and exposure falls as the loan amortizes. Early default is the expensive kind. Per our default definition analysis, the definition of the event matters too, but even with a fixed definition, its timing changes the economics.
The data a fixed window throws away
A lender has originated 1,000 loans a month for three years. To build a model of default within 24 months, it can use only loans that have been on the books for 24 months: the 12 oldest monthly vintages, 12,000 of its 36,000 loans.
- Two-thirds of the data is discarded. Loans observed for 6 or 18 months carry real information about early performance that the binary model cannot use.
- The newest information is the first excluded. The model is trained on borrowers approved two to three years ago, under older policies and conditions. Per our population stability analysis, that is exactly the drift a model should be protected from.
- Treating short histories as good is worse. Including the recent loans and labeling them non-default because they have not defaulted yet biases the model toward optimism.
Survival models handle this through censoring. A loan observed for 8 months without default contributes the information that it survived 8 months, no more and no less. Kaplan and Meier's estimator does this for a whole population, and Cox's proportional hazards model extends it to borrower characteristics, so every loan contributes for as long as it has been observed.
Hazards in practice
The central quantity is the hazard: the probability of default in a given month among loans still active at its start. A constant monthly hazard of 0.25% implies a 24-month cumulative default rate of 5.83%. Real hazards are rarely constant; consumer installment loans typically show default rising over the first months and then falling as the riskiest borrowers leave the pool.
- Discrete-time hazard models are logistic regression in disguise. Shumway showed that a hazard model can be estimated by expanding the data to one row per loan per month on book and fitting a logistic regression on whether default occurred that month. A 36,000-loan book averaging 18 months observed becomes about 648,000 rows: large, but routine.
- Time-varying inputs fit naturally. Each monthly row can carry that month's unemployment rate, the borrower's current utilization or recent payment behavior. Bellotti and Crook found that adding macroeconomic variables to survival models of credit card default improved forecasts, and per our macro overlay analysis, it lets the model respond to conditions rather than having them bolted on afterward.
- Any horizon follows from one model. The 6-, 12- and 24-month default probabilities, and lifetime loss, are all products of the same hazard curve. Per our loss forecasting analysis, that consistency is hard to achieve with separate binary models for each horizon.
Model when, not just whether
HL Hunt AI Underwriting builds monthly hazard models that use every loan on the book, including recent vintages, and produce default probabilities at any horizon from one consistent curve for pricing, limits and loss forecasting.
Prepayment is a competing risk
A loan can end in default, or it can end because the borrower repaid early. Treating prepaid loans as simply censored assumes they would have defaulted at the same rate as those that stayed, which is rarely true: the borrowers who refinance or repay early are often the ones whose finances improved. Competing-risks methods, including the subdistribution model of Fine and Gray, estimate the probability of default in a world where prepayment also removes loans from risk.
The practical consequence is in profitability, not just risk. A segment with low default and heavy early prepayment may earn less interest than its default rate suggests. Per our renewal analysis and re-default analysis, the same structure applies to lines that close, accounts that cure and modified loans that fail again: several possible endings, each with its own timing.
Lifetime losses need timing
Accounting has moved toward exactly what survival models produce. Under the current expected credit loss standard in U.S. GAAP, adopted by the Financial Accounting Standards Board in 2016, lenders reserve for expected losses over the contractual life of a loan from the day it is booked. Lifetime expected loss needs, for each future month, the probability the loan is still active, the probability it defaults that month, and the exposure if it does. A 12-month binary score supplies none of those directly, and stretching it to a lifetime estimate requires assumptions that a hazard model makes explicit.
The case for staying with binary models
- Ranking is often all a decision needs. For an approve-or-decline cutoff, a binary model and a hazard model frequently rank applicants similarly, and the binary model is simpler to build, explain and validate.
- Proportional hazards may not hold. The Cox model assumes a characteristic multiplies risk equally at every month on book. In credit it often does not; some factors matter early and fade. Discrete-time models relax this, at the cost of more parameters.
- Machine learning has binary roots. Most gradient boosting and neural network tooling is built for classification. Survival extensions exist but are less mature and less familiar to validators.
- Explanation is harder. Per our model validation analysis, a model must be explainable to validators and, through adverse action reasons, to applicants; a hazard curve is less intuitive than a single probability.
These argue for using each where it fits, not for ignoring timing. A binary model can make the accept decision while a hazard model sets price, limits and reserves, and the testable claim is simple: on a lender's own back book, compare profitability by segment predicted from a binary score with that predicted from a hazard model, and see which one matches realized results. Per our point-in-time data analysis and rare events analysis, the comparison must use data as it was known at each decision and calibrated probabilities, or it will flatter whichever model was built with hindsight.
Reserves and prices from the same curve
HL Hunt AI Underwriting handles censoring and prepayment as competing risks and supplies the monthly default and survival probabilities lifetime loss reserving requires, with documentation built for model validation.
Frequently asked questions
Methods that model when default happens, using monthly hazards and censoring so every loan contributes for as long as it is observed.
Loss depends on payments received before default. In the example, early default loses 3.9 times as much at the same default rate.
Censoring means a loan was observed for less than the full window. Survival models use such loans; binary models discard or mislabel them.
As a competing risk, since prepayers often differ from those who stay. Fine and Gray models estimate default with prepayment present.
Key takeaways
- Binary scorecards ask whether default happens in a window; survival models ask when.
- In the example, same-rate segments lose $252,921 or $64,603 per 1,000 loans depending on default timing.
- A 24-month binary model on three years of monthly vintages can use only a third of the loans.
- Censoring lets survival models use every loan for as long as it has been observed.
- Discrete-time hazard models are logistic regressions on one row per loan per month.
- Prepayment is a competing risk, not ordinary censoring.
- Lifetime expected loss reserving needs monthly default and survival probabilities.
- Binary models remain useful for ranking; hazard models are better for price, limits and reserves.
References
- Kaplan, E. L., & Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457–481.
- Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society, Series B, 34(2), 187–220.
- Banasik, J., Crook, J. N., & Thomas, L. C. (1999). Not If But When Will Borrowers Default. Journal of the Operational Research Society, 50(12), 1185–1190.
- Fine, J. P., & Gray, R. J. (1999). A Proportional Hazards Model for the Subdistribution of a Competing Risk. Journal of the American Statistical Association, 94(446), 496–509.
- Shumway, T. (2001). Forecasting Bankruptcy More Accurately: A Simple Hazard Model. Journal of Business, 74(1), 101–124.
- Bellotti, T., & Crook, J. (2009). Credit Scoring with Macroeconomic Variables Using Survival Analysis. Journal of the Operational Research Society, 60(12), 1699–1707.
- Financial Accounting Standards Board (2016). Accounting Standards Update 2016-13, Financial Instruments: Credit Losses (Topic 326).
This guide is educational and does not constitute model risk, accounting or regulatory advice. Loan terms, default timing, default rates, hazards and portfolio sizes are stylized illustrations and do not describe any particular lender or model. Losses shown exclude recoveries and funding costs. Validate any model against realized outcomes and consult accounting advisors on reserving.