Housing Aid: The APL Formula and the K Coefficient

By the end you can distinguish the formula amount from the amount after subsequent adjustments, and identify the reading used for K.

A new housing card

The Garniers rent an ordinary house. Their ALF calculation is not the formula for a logement-foyer — a collective residence. We switch to a fictional retired couple in a general-category residence autonomy, zone II, no dependants, paying €495 a month, with annual resources of €12,500. The legal month is March 2026.

Predict: can rounding K upward instead of downward change Af?

The text

Article D. 832-24 CCH, in the pinned edition applicable in March 2026, gives Af = K × (E−E0). Coefficient de prise en charge → share covered by aid → coefficient_prise_en_charge; Équivalence de loyer et de charges locatives minimale → minimum rent-and-charges equivalent → equivalence_loyer_minimale. E is the eligible rent-and-charges equivalent after applying its ceiling; in this case E = €495. Thus Af = 0.52 × (€495 − €218.12546) = €143.9747608.

Article D. 832-25 describes K as:

arrondi à deux décimales par défaut

English gloss: rounded down to two decimal places. The formula itself is published as an image. This model uses the text of the repealed R. 351-61 and R. 351-61-1 to read that formula, retaining both source references. That is a working reading; it is not a verification that the image and the earlier text have identical content.

The model

rule CoefficientPriseEnChargeAPLGeneral(m: Menage, parts: Decimal, ressources: Money) strict {
    when secteur_logement_foyer(m) and aide_demandee(m, APL)
        and type_logement_foyer(m, GENERAL)
        and nombre_parts(m, parts) and ressources_retenues(m, ressources);
    then coefficient_prise_en_charge(m, coefficient_k_general(ressources, parts));
}

Labels and source anchors are left out. The general-category function follows the working formula K = 0.95 − (R − r × N)/(cm × N), with the statutory downward rounding and cap. The household supplies facts, while the canon supplies dated parameters and operations.

Recorded on 11 October 2026 with law 0.1.12. N and K are checked in the proof of the formula answer. These are calculated components, not a claim that the fund transfers every displayed decimal to a bank account.

Change one thing

A neighbouring query proposes €146.7435062, corresponding to K = 0.53 rounded upward. Rounding this case’s unrounded K to the nearest hundredth would also give 0.52. The proposed amount is not established. It is a comparison query, not a second run adopting a different K reading. The displayed wording specifies the downward direction; the model does not replace it with nearest rounding.

What this gives the fund

A reviewer can follow each component separately: the parts, the coefficient, the minimum equivalent, the formula amount and the later adjustments. A question about K can be investigated at that step without confusing it with a CRDS deduction or a bank-payment presentation.

Where Catala stands on this

Catala’s exact decimal arithmetic is well suited to a statutory formula, and its money conversion has a defined cent rounding (Catala book, ch. 5.2). In Arxo, the rule, dated parameters and pinned source anchors form part of the executable answer. Catala 1.3.0 has no dedicated built-in form binding the result to a pinned publication’s hash; the team can maintain that archive in its application. Here the fund can inspect the source of K and the earlier-text working reading beside the component it produced. We do not compare numerical results from another implementation in this chapter.

Record and inspect

These commands require a local repository checkout and an installed law executable. Start at the repository root.

cd corpus/laws/fr/apl/examples/foyer-coefficient-k
law ask . --case CoupleResidenceAutonomie --query-json queries/aide-formule.json --out "$(mktemp -d)"
python3 check.py --law "$(command -v law)"

The local case package includes the five saved questions and a hand computation. Its check inspects N, K, applied rules and the formula’s source anchors, then executes the declared test twin.

Exercise

1. Which direction does par défaut specify?
2. Is Af the amount after all adjustments?
3. Why does the model retain the repealed articles as well as D. 832-25?