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Working note 02, , Mahé Chadeffaud and Hugo Marotin

Payment delay of public buyers: survival analysis under censoring

Abstract

The delay between delivery and payment decides how long a contract stays financed. A naive estimate takes the paid contracts and averages their delays. That estimate ignores contracts still unpaid at the time of observation, and it understates delays as a result. We treat payment as a time-to-event problem, estimate the survival curve with the Kaplan-Meier estimator, check the shape of the tail with a quantile plot, and measure the bias of the naive estimate. All figures use simulated contracts with a fixed seed.

1. Payment as a time to event

For each delivered contract we record the number of days from acceptance to payment. A contract still unpaid when the data are read is censored: we know its delay exceeds the time already elapsed, and nothing more. The Kaplan-Meier estimator uses that partial information. With d_i payments on day t_i among n_i contracts still unpaid just before t_i,

S(t) = ∏ over t_i ≤ t of (1 − d_i / n_i)

S(t) is the probability that a contract is still unpaid t days after delivery. Its median is the first day where S falls to one half.

PAYMENT SURVIVALPayment survival · 1,800 simulated contractsall contractslowest-risk 80%highest-risk 20%0501001500255075100days after deliveryP(still unpaid) %median 23 days · riskiest fifth 41 daysunpaid at day 150: 10% of the riskiest fifth · Kaplan-Meier · simulated, not live results
Figure 1. Kaplan-Meier curves of the payment delay for all contracts and for two groups ranked by the selection model.

The selection model ranks buyers before the bid (working note 03). The riskiest fifth pays later, and a larger share of it is still unpaid after 150 days. Those lots carry a higher financing cost in the bid price, or we do not bid.

2. Shape of the tail

Financing is sized on long delays, so the tail matters more than the mean. If delays were log-normal, the logarithm of the delay would be normal, and its quantiles would follow the identity line against normal quantiles.

QQ TAIL PLOTQQ plot · log payment delay vs normalnormal (identity)±0.2σ envelope-3-2-10123-3-2-10123normal quantileempirical quantile (σ)7 of 200 points beyond the envelope1,746 paid contracts · points in red sit beyond identity ± 0.2σ · simulated, not live results
Figure 2. Quantile plot of the standardised log delay against the normal distribution. Points in red sit beyond identity ± 0.2σ.

The body follows the line and both tails leave it. On the left, the shortest delays are longer than a log-normal law predicts: few buyers pay within days of acceptance. On the right, the longest delays are longer than predicted, because some contracts wait on a dispute or on a budget allocation. A single log-normal law fitted to all buyers would understate the longest delays, so we plan on the empirical 90th percentile of each buyer group.

3. The bias of dropping unpaid contracts

We draw delivery dates over observation windows from 60 to 360 days and estimate the median delay twice: from paid contracts only, and with Kaplan-Meier.

CENSORING BIASMedian delay · naive vs Kaplan-MeierKaplan-Meierpaid contracts onlytrue median60120180240300360010203040days of data observedestimated median delay (days)at 90 days: naive 18 · KM 24 · true 23 daysopen contracts are censored, not dropped · simulated, not live results
Figure 3. Median delay estimated from paid contracts only and with Kaplan-Meier, against the length of data observed.

The paid-only estimate sits below the true median on every window, by about a third on a 60-day window and still by about two days after a year, because the contracts it drops are the ones with the longest delays. Kaplan-Meier stays within a day of the true median throughout.

4. Data

On live data, the delivery date comes from the acceptance documents published with the contract, and the payment date from public treasury payment records linked to the contract by its number and date.