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

### Appendix C - Worked Example of Reimbursement Calculations

The objective of this Appendix is to demonstrate how the principles of reimbursement described above are translated into practical calculations of reimbursement payments. Initially, the calculations are illustrated using hypothetical reimbursement parameters, to help make the arithmetic more easily understood. They are then repeated with an illustrative set of reimbursement parameters taken from the analysis reported in the main text.

For simplicity, the illustrations use round numbers, and assume that 100 million concessionary bus passengers are carried at zero fare in a given period. Suppose that on average, the adult cash single fare that would have been paid on each of these journeys would have been £2.00, and that this is the value recorded as the Shadow Fare for each journey.

**Hypothetical Reimbursement Parameters**

As an initial illustration, suppose that the discount factor is 20%. This implies that the Average Fare Forgone is (100% - 20%), or 80% of the Shadow Fare. If the latter is £2.00, then the Average Fare Forgone will therefore be 80% of £2.00 or £1.60 per passenger journey.

The Reimbursement Factor is calculated from the combination of the Average Fare Forgone and the elasticity parameters (the Lambda and Beta values). Leaving the precise values of these and the algebra associated with the Reimbursement Factor to one side, suppose that, in combination, these give rise to a Reimbursement Factor of 60%. This implies that of the 100 million observed concessionary passenger journeys made at zero fare, 60% would continue to be made if passholders had to pay the Average Fare Forgone of £1.60 per journey. So 60% of 100 million, or 60 million concessionary journeys are regarded as non-generated, in other words they would continue to be made in the absence of the concession.

Hence the revenue forgone - the revenue that operators would have received in the absence of the concession - is £1.60 * 60 million which equals £96 million.

If 60% of the observed concessionary journeys are non-generated, the balance of 40% or 40 million journeys are regarded as generated by the concession. They are journeys that would not have been made unless free travel was allowed, and operators are entitled to reimbursement for the additional costs of carrying them. Assume that the additional cost rate per generated passenger is £0.40. Then the reimbursement for additional costs will therefore be £0.40 * 40 million, which equals £16 million.

Total reimbursement is therefore £96 million + £16 million, which equals £112 million.

An equivalent way of calculating the amount of reimbursement due is in terms of a Net Reimbursement Rate. The above calculations could be carried out for an individual observed concessionary journey relative to the average Shadow Fare, but doing so is counter-intuitive because it requires thinking in terms of fractions of journeys. It is easier to calculate a Net Reimbursement Rate as the ratio of total reimbursement, to the value of travel measured by the Shadow Fare. The total value of travel is £2.00 * 100 million journeys, which equals £200 million. Since Reimbursement is calculated to be £112 million, the Net Reimbursement Rate is £112 m /£200 m which equals 0.56 or 56.0%. So reimbursement can be calculated in terms of 56% of the shadow fare per observed concessionary journey.

**Calculation with Compromise Reimbursement Parameters**

Similar calculations can be carried out with any of the sets of reimbursement parameters developed within this report. For illustration here, we use the Minnerva/MVA Compromise values shown in Table 6.3 (which summarises identical calculations to those described below). The precise reimbursement parameters, some of which are year-dependent in which case we have used predicted 2012-13 values, are:

- Discount Factor: 18.60%
- Elasticity Lambda Parameter: 0.688
- Elasticity Beta Parameter: -0.471
- Price Index for Reimbursement Factor Calculations: RPI with 22% Petrol weighting which in 2012-13 is predicted to stand at 149.9 relative to 2001-2 = 100;
- Additional Cost Rate = £0.443 per generated passenger in 2009-10 prices, and £0.518 per generated passenger in 2012-13 prices.

The calculations continue to use the assumed 100 million concessionary journeys, and an average shadow fare of £2.00.

If the Discount Factor is 18.60%, then the Average Fare Forgone is (100% - 18.60%) or 81.40% of the shadow fare of £2.00, or £1.628. This is the fare revenue that the bus operator would have received for each journey made by passholders if the concession didn't exist i.e. that is not generated.

In order to apply the elasticity parameters, it is necessary to convert the Average Fare Foregone from current prices (assumed to be 2012-13) to 2001-2 prices - the price level of the fares from which the elasticity parameters were determined. If the weighted price index in 2012-13 is 149.9, it implies that the relevant prices were 49.9% higher than in 2001-2, and that the 2001-2 price equivalent of the 2012-13 Average Fare Forgone of £1.628 is (£1.628/1.499) which equals £1.086.

The Reimbursement Factor is calculated from the formula RF = Exp(Beta * Fare ^Lambda), or Exp(-0.471 * £1.086^{0.688)}. Calculating this in parts, £1.086 raised to the power of 0.688 = 1.0584. This value multiplied by Beta is 1.0584 * -0.471 which equals -0.4985. Finally, the exponential constant e is raised to this power, which equals e^{-0.4985} or 0.6074. The Reimbursement Factor is therefore 60.7%, in other words it is estimated that if the average fare forgone was £1.086 in 2001-2 prices, then of the 100 million concessionary journeys observed, 60.7 million would continue to be made in the counterfactual, and 39.3 million concessionary journeys are regarded as generated by free travel.

Since in the counterfactual, the bus operator would have received an average fare forgone of £1.628, the total revenue forgone from the 60.7 million non-generated journeys will be 60.7 * £1.628 = £98.8 million.

The bus operator is not entitled to receive any revenue forgone from the 39.3 million generated journeys, but is entitled to reimbursement for the additional costs incurred in carrying them. With the additional cost rate of £0.518 per generated passenger, the additional cost reimbursement will be 39.3 million * £0.518 which equals £20.36 million.

Total reimbursement is therefore £98.8 million + £20.36 million, which equals £119.2 million.

Another way of expressing the result of the calculation is as a Net Reimbursement Rate per £ of shadow fare. This is how the current Scottish system is applied. The Net Reimbursement Rate can be calculated from £119.2 million, divided by what could be called "the value of travel" - the product of the observed concessionary journeys, and the shadow fare, in other words £119.2 / (£2.00 * 100) = 119.2/200 = 59.6%.

The Net Reimbursement Rate could therefore be thought of in terms of the proportion of the Shadow Fare that an operator is paid for - so a concessionary passenger carried at a shadow fare of £1 receives reimbursement of £0.596, or at a shadow fare of £2.00, reimbursement of £1.192.

The above calculations could all be carried out on a per-observed concessionary passenger basis, but is more intuitive to calculate it in terms of reimbursement for a given volume of concessionary journeys, so that concepts such as reimbursement factors do not imply fractional (non-integer) journeys.