**Time and Discounting**

[2500 Words; 20 Minute Read] For major project feasibility and decision making, if cost and benefit items have been identified and valued, the net difference of costs and benefits need to be considered in relation to time. The reason that time is factored into a particular project is to account for the change in the value of money over the project period. The value of money changes over time due to fundamental principles of how people will prefer to satisfy want now rather than later. Conversely people will also typically prefer costs later rather than now. As an example satisfaction of wants such as buying a house will be preferred now (from a consumer point of view), although the costs of paying for the house would be preferred sometime into the future or even deferred to the point of non-payment (if the seller allows it). This difference in satisfaction now and incurred cost at a later date, mean that goods and services will typically have a higher net benefit and cost in the future, and thus this change in future increased good or service’s value needs to be discounted back to today’s price of money. Or put differently, because individuals attach less weight to a benefit or cost of a project in the future than they do to a benefit or cost now (because the transaction is important to the buyer and seller at today’s monetary value), the monetary value will have to be adjusted (or discounted) to more accurately depict the true net cost or benefit at time periods ranging from the start of the project to the end of the project.

To apply an example, the future benefit of infrastructure in year 1 may be valued at £100M and in 20 years may be valued at £10M, but this value in year 20 the net benefit of £10M is using the cost of money 20 years ago so needs to be adjusted. The discount rate operates like the opposite to interest if money was saved rather than invested, for instance if money (M) is put into the bank (rather than invested in infrastructure) at an interest at 10% over 20 years the value of money in 20 years would be worth Mx20x10%. To counteract for an opportunity cost in savings (% interest received from the bank) foregone when investing (in infrastructure) the discount rate of money needs to be applied for each period of time.

This method of discounting can consider the ‘real’ net cost or benefit of goods and services over time – but it can also be used to consider the environmental impact of project over many years that stretch over several generations (e.g. 100 years). For instance the net benefit of a pollution control initiative can be valued into the future whilst calculating how the changing value of money will affect the net benefit at certain time periods. Note that the discounting technique is simply to generate estimate valuations into future periods, and does not necessarily reduce environmental costs in the future. For instance the changing of discount rates can be used to alter the amount of net benefit or cost will be incurred towards the start of the project (say in year 5) or at the end of the project (say in year 20). If there was a reduction in the discount rate from 10% to 5% this would mean that the net benefit or cost is discounted more in the earlier years and would have less consideration for future net benefits or cost. As the inverse, an increase in discount rate from 5% to 10% would mean that net benefit or cost would be discounted less in earlier years and have more consideration for future net benefits or cost. An more in depth example is applied shortly but what this means is this, if higher discount rates (e.g. 10% rather than 5%) are applied to a project that has high environmental impact costs in the future, the ‘real’ monetary costs will of these high future costs will have been considered and dampened in the model – to account for how money will be worth less in the future. To round of this initial discussion in lay terms, discounting occurs as people would prefer to have a dollar in there hand today rather than in the future because the dollar will be worth less in the future. This in turn means that if the dollar is worth less in the future, any project in the future that has benefits or costs, needs to be adjusted account for such devaluation of money over time.

**The Mathematics of Discounting**

Now that discounting has been introduced to valued net benefit or cost, the mathematics of how a discount rate is applied should be briefly demonstrated. From Table 1, over a five year period the aggregated net benefit has been calculated from year 1 to year 5 as valued in currency (£,$,€ etc) of -30 (a net cost), -5, 15 (a net benefit), 15, and 15. This means that as the 5 year project (e.g. an urban by-pass) progressed it went from a heavy initial cost to become a sustainable net benefit from year 3 to year 5.

Figure 1: The formula used for applying a discount rate to a net benefit or cost

Table 1: Table of aggregated values for discounting: benefit, costs, and net benefit

A discount rate is then calculated to the net benefits and costs for each year. As an example here, a discount rate of 10% is used that is equal to a decimal of 0.1. In relation to the formula is Figure 1, this means that the part of the formula that uses 1 + r (where r is the rate of discounting) the value to be used in each calculation is 1 +r, which equals 1 + 0.1, or if added together 1.1. From the formula in figure, t represents time and the sigma sign represents the summation of all 5 years of calculation. Hence, when plugging in the numbers into a long-hand equation, the calculation that should be processed is as follows:

From this calculation it is found, as to be expected, that the final Net Value (NVs), which is the total net benefit or cost before discounting, is different to the final discounted NPV. In this example, the figures gave a NV of 10 (Benefits – Costs = 45 – 35 = 10) without applying a discount. Although when applying a discount, the final discounted NPV was at negative 5. What this figure shows is that a negative total project value means that the project should not go ahead. Especially as all costs and benefits over time, and changes in monetary value over this time have been considered. Despite this considerations could be made in practice as to whether this 0.5 value is of significant material value to not go through with the project, or alternatively different discounted rates could be applied to ensure there is a positive net benefit – but most importantly justifying the discount rate is a true and fair one.

**NPV (Net Present Value) and Discounted values for whole project**

The short 5 year introductory CBA example for the purposes of mathematical understanding can now be transposed to a more practical project template. Table 2 is one such CBA template example that considers: the costs; benefits; net benefit or cost; and 2 alternative discount rates (5% and 10%) over a 15 year period. Here as per step 1, cost and benefit items are identified such as the actual construction and maintenance costs, and the benefits generated from shorter journey times. The second step of applying monetary values are put into the CBA with say 70K being applied to the third year of maintenance costs. The third step of adding together all costs and benefits are then made, with the addition of another row of calculation that is the Net Value (NV) of total costs taken away from total benefits. The forth step in the process is discounting, and here both a 10% rate and a 5% rate are applied in order to show how the variance in rate will influence the weight in calculation of the present values, and subsequently adjust the final total discounted NPV. In the template provided it can be seen that a lower discount rate (5%) has less of an influence in reducing the NPVs (e.g. in year 1 multiplying through by 0.95 rather than 0.91, means that a multiplication closer to 1 doesn’t reduce the net benefit or cost by a greater amount). The important figure at the end of the analysis is the value in the bottom right hand corner that is the total discounted NPV (Table 2: *L-F). If this value is positive it indicates that according to the figures used the particular project should go ahead. If the value is negative the opposite is true and the project should not go ahead. As mentioned in the previous section the discounted value can be adjusted in the spreadsheet (if using more sophisticated tools away from first principles demonstrated here for teaching purposes) in order to generate a positive discounted NPV, although justification of the value will be required for a credible analysis to be approved.

Table 2: CBA Template for an Urban Bypass Proposal

**A Simple Discounting Example: Cost of Nuclear Waste**

Whilst bearing in mind these technical steps in producing a CBA, another basic scenario that will make this analysis clearer is in the costing of nuclear waste disposal that has been used to generate power for urbanizing spaces. Here, in the CBA of building a nuclear reactor the discounting applied to the Net Value (NV) figures will mean that the future NPV will change in value depending on the discount rate used. If say a low discount rate was applied this would mean that future costs and benefits of installing a nuclear reactor would have (on paper) higher net cost or benefit in the future. If this future value was a cost that spanned many generations, for instance, if the waste disposal costs were higher than a reactor not generating any power or jobs, the discount rate applied will make a large difference as to whether the future cost is high enough to discourage a decision to build the reactor now. Figures will now demonstrate how discounting can shift costs to future generations. Particularly, if nuclear waste dumps from the past can produce leakage costs to future generations.

Say if the cost of leakage is £1 billion (1000 million) in 100 years time, the value (NPV) of the £1Bn in monetary terms will be far less if the value of money is worth less. To calculate what £1Bn is worth now (NPV) to make a decision on whether the project should go ahead now a discount rate of 8% (or as a decimal 0.08) is applied. Using the formula in Figure 1, and ‘plugging in’ the numbers, the following result of £450,000 is calculated as follows:

This means that the damage costs in 100 years at today’s price of money is £1 Bn, but at prices in 100 years time the value of costs is only at £450,000. From this it can be seen that the application of ‘**time’** (particularly the time value of money) is a key-determining factor of analysing future costs and benefits through discounting.

**Cost Benefit Analysis (CBA) use in Practice**

The future net costs or benefits that change through discounting application are a paper analytical exercise to aid decision making on whether a project should go ahead or not. In practice, more factors need to be considered beyond the number crunching to realise what the number can represent and how they will be biased to meet different interested parties in a project. For instance, both public and private sectors use NPV (Net Present Value) in project appraisals, although CBA itself is primarily used for project appraisal in the public sector – as it considers public goods and services as well as private ones. For the private sector, the use of financial appraisal or capital budgeting is often more appropriate rather than a CBA that considers external costs and benefits. In the public sector, these further external costs and benefits will use value not determined directly by the market, and use values by alternative methods as discussed above such as the Willingness to Pay (WTP) principle or the opportunity costs (forgone opportunities – the next best alternative) should a project go ahead (e.g. build a nuclear reactor) over another project (e.g. build a school).

**Summary**

- In major projects, if cost and benefit items have been identified and valued, the net difference need to considered in relation to time. The reason that time is factored into a CBA of a particular project is to account for the change in the value of money over the project period. Each cost and benefit can be discounted (which can be conceptualised as the inverse of an interest rate) for each year of application according to the percentage rate applied.
- The monetary value will have to be adjusted (or discounted) to more accurately depict the true net cost or benefit at time periods ranging from the start of the project to the end of the project.

- The Net Present Value (NPV ) is the net figure for costs minus benefits that have been discounted for each year in operation. For the building a nuclear reactor the discounting applied to the Net Value (NV) figures will mean that the future NPV will change in value depending on the discount rate used. The future generational cost in say 100 years can therefore be factored into the model to provide a present value that incorporates the future cost (and benefit)
- The future net costs or benefits that change through discounting application are a paper analytical exercise to aid decision making on whether a project should go ahead or not. In practice, more factors need to be considered beyond the number crunching to realise what the number can represent and how they will be biased to meet different interested parties in a project.
- For the private sector the use of financial appraisal or capital budgeting is often more appropriate rather than a CBA
- In public sector CBA, these further external costs and benefits will use value not determined directly by the market, and use values by alternative methods as discussed above such as the Willingness to Pay (WTP) principle or the opportunity costs (forgone opportunities – the next best alternative) should a project go ahead (e.g. build a nuclear reactor) over another project (e.g. build a school).

A full and formated version of this article can be cited as:

Squires, G. (2013). Chapter 8: Cost-Benefit Analysis and Discounting. In Squires, G. (2013) ‘Urban and Environmental Economics. Routledge.