Existing invoices and new sales move on different terms

Consider a separate 60-day transition. Opening unpaid invoices are 60: 40 falls due on day 15 and is expected to produce 38 then; the other 20 is already overdue, with expected collection of 10 on day 45. The proposed policy does not change these invoices or their expected losses. Without the policy, new sales of 100 occur on day 0 and again on day 30, each payable 30 days later. Expected collection is 95% of each invoice, giving 95 on days 30 and 60.

For new sales only, customers accept an offer of a 2% discount for payment 15 days after invoicing. The supported operating scenario raises each new sales cohort from 100 to 120; the expected paying share remains 95%, with the other 5% producing no receipts within or after this comparison. Thus each changed cohort produces 120 × 0.98 × 0.95 = 111.72 on days 15 and 45. Feasible delivery requires cash equal to 70% of the undiscounted invoice amount on the invoice date: 70 per cohort before the change and 84 after it. There are no other costs, taxes or remaining operating differences. Opening usable cash is 80 and the stipulated minimum balance is zero.

Construct both accounts, including the unchanged opening invoices:

On day 30 the no-change account receives 95 from its first new cohort and pays 70 for its second. The changed account has already collected its first cohort and pays 84 for the second. Opening-invoice collections cancel in the incremental column because their terms and performance have not changed; they still belong in each absolute cash account.

The no-change cash path is 10, 48, 73, 83 and 178. The changed path is −4, 145.72, 61.72, 183.44 and 183.44 before any new finance. Earlier collection reduces later receivable funding, but the larger first delivery needs at least 4 of obtainable net finance immediately. Future expected receipts cannot make that payment now. FIN.2 must add the actual financing terms and test the relevant adverse collection cases before the changed policy can be relied on.

Final expected cash improves by 5.44, not just by the margin on the new sales. For each cohort, the additional 20 of sales contributes 20 × 0.98 × 0.95 − 14 = 4.62; granting the discount on the existing 100-sales base sacrifices 100 × 0.02 × 0.95 = 1.90 of expected receipts. Twice their difference is 5.44. Expected noncollection is already in these receipts and is not another expense to subtract from cash.

At a qualified 1% effective rate per 30 days for the specified expected incremental flows, their present value is 6.18, using the corresponding half-period factor for days 15 and 45. If the offer produces no additional sales, the volume and cost remain 100 and 70 per cohort, while discounted expected collection becomes 93.10. The value difference is then −2.83 on the same basis. The commercial response changes the preference; holding quantities equal merely to make the alternatives look comparable would lose the question.

The old overdue 20 remains identifiable in the aging until actual settlement or the applicable write-off treatment changes it. Growing recent sales can improve an aggregate days-receivable measure without collecting any of that overdue balance.