# Scotland-wide Older and Disabled Persons Concessionary Bus Scheme - Further Reimbursement Research

### Appendix A Overview of bus reimbursement calculation principles

In principle, the calculation of "no better off, no worse off" reimbursement involves a number of distinct elements, although in practice these are often combined or simplified. The Scottish reimbursement formula reduces the components into a single factor, nominally calculated to reflect average all-Scotland characteristics, applied to an operator's average adult single fare, but implicitly it incorporates assumptions about each of the different elements. Figure 1 summarises how each of the different elements interact in order to calculate reimbursement of a public transport operator for providing free concessionary travel.

**Figure 1 Reimbursement Calculation Flow Chart**

The calculation process involves a number of steps.

**First**, the average fare that would be paid by concessionary passholders in the absence of the concession (the "commercial" fare or Average Fare Forgone) is calculated. In the DfT Calculator, the calculation is driven by an estimate of the average cash fare paid, which is then reduced via a discount factor intended to reflect the availability of various discount tickets (such as day tickets), and the probability of these tickets being used by passholders.

**Second**, a demand model (which simulates how passenger volumes vary with changes in fare) is used to determine the proportion of concessionary passenger journeys "generated" by the concession. This is necessary because it is not possible to directly observe the journeys that would have been made in the counterfactual. Instead, observed concessionary journeys are used as a proxy, but with allowance made for generation, i.e. the proportion of concessionary journeys that would not have been made at the commercial fare.

In the DfT Calculator, the measure of generation is the Reimbursement Factor, which is the ratio of passenger journeys at the commercial fare to passenger journeys at the concessionary fare. The proportion of travel "generated" by the concession is often known as "the Generation Factor" but at least two alternative definitions are in common use, and the Reimbursement Factor (the ratio of non-generated to concessionary journeys) is the preferred measure of generation as it provide less scope for confusion.

The demand model parameters reflect assumptions about the sensitivity of demand to fare levels, often summarised as an **elasticity**: see note below. The Reimbursement Factor is a function of these parameters, as well as the average far forgone that concessionary passengers would have paid in the counter-factual.

Third, once the Reimbursement Factor has been calculated, it can be applied to the observed quantity of concessionary journeys to separate non-generated (i.e. counter-factual) journeys from generated journeys.

**Fourth**, since the objective is that the operator should be financially no worse off, the reimbursement due for revenue forgone is the revenue that would be earned by the operator ie the hypothesised commercial revenue. This is the product of the non-generated ("commercial") trips and the commercial fare.

**Fifth**, the fact that the operator will carry additional passengers because of the concession may mean that it incurs additional operating costs. The DfT Calculator estimates an additional cost rate per generated passenger using various factors that reflect network characteristics, which when multiplied by the number of generated passengers gives reimbursement for additional costs.

Total "no better off, no worse off" reimbursement is then the sum of revenue forgone and additional costs.

In the Scotland-wide Free Bus Travel Scheme for older and disabled people, a single factor which is termed the Net Reimbursement Rate is applied to the adult cash single fare that would be charged for each concessionary journey made. The rate implicitly includes allowance for all three of the distinct concepts of:

- the reimbursement factor (allowance for generation);
- the discount factor (allowance for the likelihood that concessionary passengers would use discounted tickets rather than the adult cash single fare, to some extent); and
- additional costs (allowance for the likelihood that additional operating costs would be incurred in carrying generated concessionary passengers).

The great advantage of a single, fixed reimbursement rate is the simplicity of the concept. It can be applied at any level of aggregation. The reimbursement owed to an operator can be calculated for an individual concessionary passenger, provided that the adult cash single fare that would have been paid for that journey is known. Or alternatively, it can be calculated for an operator, for the total number of concessionary journeys carried, or for all operators, provided that the average value is known of the adult cash single (Shadow Fare) fare that would have been paid for the journeys being reimbursed.

Application of a single Reimbursement Rate at any level gives an arithmetically identical result. However, concepts such as generation, discount factors and additional costs are hard to visualise in terms of a fraction of the adult cash single fare of an individual concessionary passenger journey. They make much more sense in terms of a proportion of a given total of concessionary journeys (as in the proportion of observed concessionary journeys that would continue to travel in the absence of the concession), although it is necessary to think in terms of the average value of concessionary journeys (eg the average adult cash single fare) rather than the fare that would be paid for an individual journey. Moreover, since reimbursement through the Scottish National Scheme has the objective that operators as a whole are left no better off and no worse off, determination of the reimbursement rate should be on the basis that it matches the average characteristics of operators as a whole.

It should be noted that irrespective of the level at which components of reimbursement are calculated, it is always possible to apply the resulting calculations using a single rate, but care is needed to ensure that appropriate mechanisms are in place to systematically update the Rate so that it continues to deliver "No Better Off, No Worse Off" reimbursement at the chosen level of aggregation.