Study · simulated data · 2026-10-09

Approving personal loans for profit when the default rate breaks

Which applications a lender approves, so that the profit of the loan book is highest, when the book's default rate drops for a year and then rises above its old level. A scorecard, a loan-by-loan rule and a PD level that follows early delinquency, on a simulated book of 72,000 applications.

10.34 M EUR

profit over the test period

12 months, 18,000 applications; recommended rule

+2.02 M EUR

against the level from the latest known year

the level a lender gets by default

71.6%

of applications approved

default rate among them 7.45%, against 12.16% of all applications

0.786 / 0.738

scorecard ROC-AUC, validation / test

boosted model 0.822 / 0.749; it earns less at every level

The study in six sentences

The business question	Which applications for a personal loan does a lender approve, so that the profit of the loan book is highest?
The decision the business takes with the answer	Approve or decline each application, at the time of application.
The rule or model that makes that decision	Approve when the predicted PD of the application is below the break-even PD of its loan, which follows from the loan's interest income, its amount, the LGD and the required margin $m$.
What it earns, in money	9.53 M EUR over the test period of 12 months (18,000 applications) with the scorecard at the loan-by-loan rule, against 0.39 M EUR for approving every application and 9.18 M EUR for the scorecard with one fixed cut-off; the boosted model earns 9.35 M EUR at the same rule (the difference to the scorecard lies inside its 95% interval). At the models' PD level, which is 5 points below the test period's default rate of 12.16%; the level options are sentence 5.
The main risk, and what is done about it	The default rate of the book changes abruptly: a payment-relief programme lowers it to 2.9% for a year, and it rises to 12.2% when the programme ends. No application shows this, and at the decision date the latest known default rate is still the programme's. The PD level is set by a monitor: the average before the programme, raised each month in proportion to the early delinquency rate of the latest three months, never lowered. With it the scorecard earns 10.31 M EUR; with the level taken from the latest known year it earns 8.33 M EUR, and from the average before the programme 9.54 M EUR. The level is reviewed every 3 months.
The recommendation	Adopt the loan-by-loan rule with the scorecard at the monitor level and a required margin of 0.02 EUR per EUR lent (10.34 M EUR, 71.6% of applications approved), after a champion-challenger test on 20% of applications for 6 months, read at month 3 on early delinquency and decided after 12 months on profit per EUR lent. The boosted model ranks better (ROC-AUC 0.749 against 0.738) and earns less at every level.

The book and the break in its default rate

The simulated book has 1,500 applications in each of 48 months. A payment-relief programme runs from month 19 to month 30: loans applied for in those months show a 12-month default rate of about 3% against 7% before. After the programme the rate rises to 12%. The decision date is the start of month 37; the rule is scored on the applications of months 37 to 48.

At the decision date a 12-month outcome is known for applications up to month 25, all of them before or inside the programme. The only sign of the period after the programme is the early delinquency rate: the share of a month's loans with a missed instalment in their first 3 months, known 3 months after application.

Figure 1. What it shows: the 12-month default rate (solid) and the early delinquency rate (dashed) of each application month, both as a share of that month's 1,500 applications, in %. Shaded: the programme months and the test period; dotted: the decision date. The horizontal lines are the four PD levels a lender could set at the decision date from what is known then, drawn across the test period. Message: the latest known default rate at the decision date is the programme's, about 3%; the early delinquency rate had already risen before the decision date, and it is the only signal of the 12% to come. Made from the simulator's outcomes over all 48 months; the test period's default rate is known here in hindsight only.

The rule

A loan is approved when its predicted probability of default (PD) is below its break-even PD:

PD* = (I − m·A) / (I + LGD·A)

I: interest income of the loan over its term if repaid, in EUR
A: loan amount, in EUR
LGD: loss given default assumed by the rule, the share of the amount lost after recoveries, ratio; 0.65 for every loan
m: required margin, the minimum expected profit per EUR lent, ratio; 0 is the risk-neutral rule

Profit over the test period is the interest income of approved loans that are repaid, minus the realised loss of approved loans that default (their own loss given default times their amount), in EUR. A defaulted loan is counted as paying no interest.

policy, at the models' own PD level of 7.1%	profit, M EUR	approved, % of applications	default rate among approved, %
approve every application	0.39	100.0	12.16
scorecard, one fixed cut-off	9.18	87.2	8.62
scorecard, break-even PD per loan	9.53	82.8	8.34
boosted model, break-even PD per loan	9.35	88.1	8.71
true PD, break-even PD per loan (ceiling for a model)	11.75	75.6	6.04
perfect foresight (approve every loan that repays)	21.17	87.8	0.00

The loan-by-loan rule earns 0.35 M EUR more than one fixed cut-off with the same scorecard: it declines the loans whose own interest does not cover their own risk. The better-ranking boosted model earns less than the scorecard: at a PD level 5 points below the test period's default rate, both models approve too many loans, and the one that ranks better approves more of them.

The PD level: four ways to set it at the decision date

The PD level is what the book's predicted PDs average to. Each model's PDs are shifted in log-odds to reach it. Four definitions, each from what is known at the decision date:

Point in time: the default rate of the latest 12 application months with a known outcome. This is the level a lender gets by taking the latest year.
Through the cycle: the default rate of the known months outside the programme.
Stressed: through the cycle plus the programme's deviation below it, mirrored upward, on the judgment that payments deferred by a programme come back as defaults after it.
Monitor: through the cycle, raised each month in proportion to the early delinquency rate of the latest 3 known months against the rate before the programme, and never lowered.

Figure 2. What it shows: profit over the test period, in M EUR, for each PD-level option (with its level in %) and each model, at the loan-by-loan rule with margin 0. Message: the level moves profit by 2.0 M EUR between the worst option and the best; the model choice moves it by 0.1 to 0.4 M EUR, with the scorecard ahead at every level. The monitor, chosen before the test period was opened, is 0.04 M EUR below the stressed level. Made from reports/metrics_levels.json.

option	level, %	scorecard: profit, M EUR	scorecard: approved, %	boosted: profit, M EUR	boosted: approved, %
point in time	4.74	8.33	88.5	8.33	92.5
through the cycle	7.11	9.54	82.8	9.35	88.0
stressed	11.35	10.35	72.7	10.22	77.3
monitor	9.22	10.31	77.8	9.96	83.1

Monitor against through the cycle, scorecard: +0.77 M EUR, 95% interval +0.45 to +1.12 M EUR over 1,000 resamples of the applications. Boosted model against scorecard at the monitor level: -0.75 to +0.03 M EUR. The monitor reaches 9.2% against a true 12.2%: early delinquency rises less than proportionally to default, so it under-reacts; a lender with its own ratio of early delinquency to default sets the proportion from it.

Risk appetite: the required margin

The bank's appetite enters the rule through one parameter, the required margin m. The recommended policy at three values:

required margin, EUR per EUR lent	profit, M EUR	approved, %	default rate among approved, %	amount lent, M EUR	profit per EUR lent, ratio
0.00	10.31	77.8	7.73	131.4	0.0785
0.02	10.34	71.6	7.45	119.8	0.0863
0.04	9.99	62.4	7.35	104.4	0.0957

At 0.02 the lender earns the same profit with 11.6 M EUR less lent, at a lower default rate. The value of m is the bank's; 0.02 is the recommendation on this data.

The scorecard

A logistic regression on binned inputs: payment burden (instalment plus existing debt payments, over net income), employment years, credit history years, missed payments in the last 24 months, loan amount, term, housing, purpose. Each numeric input is cut at the quintiles of the fitting months; a missing income is its own bin of the payment burden. Net income and existing debt enter only through the burden: beside the ratio built from them they had no reading of their own, and the burden's bins came out of order.

Figure 3. What it shows: the contribution of each payment-burden bin to the score, in log-odds; bins to the right raise the predicted PD. Message: the burden's bins rise in order, and the step above 0.378 is the largest single contribution in the scorecard, so a lender can read the line and act on it. Made from reports/scorecard.csv.

The boosted model (gradient-boosted trees on the same inputs; 15 leaves, 200 trees, chosen among 36 configurations on the validation months) ranks better: ROC-AUC 0.749 against 0.738 on the test period. At these levels the better ranking is worth nothing in profit, and the scorecard is the form a lender adopts directly.

The production test

Champion-challenger: the recommended rule decides a random 20% of applications for 6 months, the current rule the other 80%. First read at month 3 on the early delinquency rate of approved loans; final decision after 12 months on profit per EUR lent. The challenger is adopted if its profit per EUR lent is higher and its early delinquency rate is not higher. At 1,500 applications a month, 20% over 6 months is 1,800 applications on the challenger, enough to see a 2-point difference in early delinquency at a rate of 9% (standard error of the difference about 1 point).

Decisions, with their cost

issue	decision	cost
which data	the loan book is simulated; the simulator is published with the study	proves the method, not a result for a real lender
the break in the default rate	a payment-relief programme: 7% before, 3% during, 12% after	its size is chosen, not measured
the programme months in the fitting data	kept for ranking through an indicator switched off at prediction; left out of the PD level	assumes the programme shifts every applicant's odds by the same factor
income missing for 2.9% of applications	its own bin of the payment burden	the burden cannot be computed for them
instalment, interest income, interest rate	left out as model inputs; they are computed by the lender from inputs the models have	none for ranking
loss given default in the rule	one assumed value, 0.65, the mean of the simulated distribution	profit understated for loans that default late
which PD-level option	the monitor, chosen before the test period was opened; all four reported	0.04 M EUR below the best option
which model	the scorecard: more profit at every level, and readable	0.011 of ROC-AUC given up
required margin	0.02 EUR per EUR lent	6.2 points of approvals

Limits

Simulated data: every number is a property of the simulator's choices, published with the code. The method transfers; the numbers do not.
Every outcome is known, including for applications a lender would decline; reject inference does not arise here.
The level is set by one shift of the log-odds, not a full recalibration; the level known in hindsight earns less than the stressed level for that reason.
The applicants carry no protected attribute, so no fairness analysis is possible on this book.
Two models and one small fixed grid; the question is the decision, not the model family.

How this was made: a plan with six sentences and a time budget written before any data existed; the test period locked and opened only by scripts that log each opening; every decision recorded with its reason and its cost; the recommendation written down before the test period was opened and reported beside the alternatives. Code: studies/lending-decision in the Hope repository, Python 3.12, seed 20261005.