This is rather odd. The
DfT» 's recommended method for calculating reimbursements doesn't use the actual journey cost or length at all, partly because it can't use whether you used some "bulk" ticket (e.g. day or week) - since a pass user would not buy one. Instead, it uses a flat rate per journey, which also allows for the way smartcards are used, i.e. without asking for a destination or fare in most cases.
Note that, as their guidance says: "
TCAs▸ are free to use the methodology of their choice in estimating reimbursement subject to ensuring compliance with European regulation No 1370/2007 as well as relevant domestic legislation that governs concessionary travel reimbursement." I have no idea whether any TCA has actually invented its own method (and of course checked its compliance with all that legislation), nor where to look to find out.
The calculation of this daily rate, so as to approximate well enough to the revenue neutral ideal, depends on good input data. These need to cover such things as what fraction of pass users would have paid the fare, or bought some other ticket, and what the journey cost distribution would have been, though here is a lot more than that. They offer some default data, suitable for allocating smartcard counts to ticket types:
Smartcard Data Ticket Choice Assignment
H.9 Smartcard data on journey frequencies from the NoWcard scheme have been used to model how concessionary passholders would allocate themselves to different ticket types (cash, daily and weekly tickets) and fares at free fares. The data provides information on the concessionary journeys of about 90,000 passholders made over a five-week period in four Lancashire districts.
H.10 The data have been summarised to give the number of concessionary journeys made in each day of the five-week period, as well as the number of journeys made in each of the five weeks. The summarised data have then been used to simulate how the observed travel patterns would map onto different ticket types, assuming different combinations of price ratios.
There's a lot more of that kind of stuff in
the guidance note. And theory, and algebra - lots of those too.