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Basics of Discounting

 

Introduction

16.1

Using a discount rate has the effect of reducing the value of future costs and benefits in present day terms. If society has a discount rate of 3.5% per annum, this implies that it values £1 today equally with the certainty of £1.035 in a year's time. Another way to express this is to say that £1 in a year's time is worth only 96.62p now, because 1/1.035 equals 0.9662. The 96.62p figure is called the Present Value (PV) of the £1, and the 0.9662 figure is the relevant 'discount factor'.

16.2

The following figures show how the PV of £1 declines in future years when the rate of discount is 3.5% per annum.

Year of Payment
(mid year)

Present Value
(at middle of year 0)

0

 £1.0000

1

 £0.9662  ( = £1 x 1/1.035)

2

 £0.9335  ( = £1 x 1/1.0352)

3

 £0.9019  ( = £1 x 1/1.0353)

10

 £0.7089  ( = £1 x 1/1.03510)

16.3

It is important to remember that the discount rate should generally be applied to figures that are:

  • expressed in real terms i.e. excluding allowance for general inflation (See 2.8.6 above); and
  • adjusted for appraisal optimism (See 2.6.15 above)
16.4In most appraisals it is sufficient to carry out discounting on costs and benefits identified at annual intervals. For example, it is common to identify streams of costs and benefits assumed to occur in the middle of Years 1, 2, 3 etc and to discount them all back to the middle of Year 0. Similarly, they may be assumed to commence at the start of Year 1, 2, 3 etc and discounted back to the start of Year 0.
16.5Table 1 shows the discount factors needed to calculate PVs at 3.5% per annum. Table 2 provides discount factors for discount rates from 1% to 10% per annum. Detailed discounting calculations are facilitated by the use of suitable computer software, avoiding the need to refer to discount tables. However, tables can be useful in some circumstances, for instance when simple calculations are required. Departmental economists can advise on the design of spreadsheets to suit particular cases.

Net Present Value
16.6Net Present Value (NPV) is the name given to the sum of the discounted benefits of an option less the sum of its discounted costs, all discounted to the same base date. Its significance as a key summary indicator of the comparative value of an option is explained in section 2.8 of the NIGEAE. A negative NPV is generally referred to as a Net Present Cost (NPC).
Example: Equipment with 4 years of life requires replacement. There are two options A and B with different capital and recurrent costs. No monetary benefits are identifiable. The aim is simply to choose the least cost solution ie that with the lower NPC.

  0 1 2 3 4
Discount Factor 1 0.9662 0.9335 0.9019 0.8714 NPC
Option A £k £k £k £k £k £k
Capital Costs 100
Revenue Costs 35 35 35 35
Present Value 100 33.8 32.7 31.6 30.5 228.6
Option B
Capital Costs 75
Revenue Costs 47 47 47 47
Present Value 75 45.4 43.9 42.4 41.0 247.7

Despite higher capital costs initially, Option A has the lower net present cost and should be preferred to Option B.

Timing Considerations
16.7NPVs should generally be calculated over the same time period for all options. The selected time horizon should reflect the economic life of the services in view, or the useful life of relevant key assets. It should be sufficiently distant to cover all the important cost and benefit differences between the options. For example, the appropriate period may be 5 years for an information technology project, 10 years for an industrial development project, or 25 years or more for a longer term project.
16.8

Assets with different lives can be accommodated within this approach by assuming replacement at appropriate times and making suitable use of residual values (RVs). For instance,

  • Suppose Option 1 involves replacement of a piece of equipment with a 15-year life in Year 10 of an appraisal, while Option 2 requires replacement of similar equipment in Year 5;
  • Using a common time horizon of 20 years, the difference in useful life remaining at the end of 20 years will be accounted for by assigning different RVs to each option in Year 20;
  • In Option 1 the equipment will be assigned an RV of approximately one third of its value, since at Year 20 it will still have 5 years i.e. one third of its useful life remaining;
  • In Option 2 the equipment will have a zero RV, because it will be at the end of its useful life.

Equivalent Annual Costs
16.9In some cases it can be helpful to calculate PVs in terms of Equivalent Annual Costs (EAC), rather than NPVs or NPCs. A cost of £100 in the middle of Year 0 is equivalent to a stream of 10 annual costs of £12.03 starting in the middle of Year 1 when using a 3.5% annual discount rate. It can be demonstrated that such a cost stream has a PV of £100 when discounting at 3.5% per annum. An asset that costs £100 and has an expected life of 10 years is thus said to have an EAC of £12.032. Table 3 below provides EAC factors for a 3.5% discount rate.
16.10EACs can be useful when contemplating replacement of a capital asset, where there is a need to compare alternative assets with different lives.

Example: Consider two options for replacing a boiler. In Option X a boiler with an expected life of 7 years may be purchased for £2,000. Under Option Y another boiler with an expected life of 10 years may be purchased for £2,500. Which should be purchased? The relevant costs may be annuitised using EACs as follows:

Option

Life (yrs)

Cost (£)

EAC Factor (@3.5%)

EAC (£)

X

7

2,000

X

0.1635

=

327

Y

10

2,500

X

0.1203

=

301
In this case, the initially more expensive boiler would be the more cost-effective choice.

16.11Table 3 contains annuity factors, which are convenient for discounting a series of constant annual costs or benefits. For instance, suppose a constant annual payment of £10,000 in real terms has to be paid every year from Year 1 to Year 20. The PV of this cost stream may be calculated by applying the annuity factor for 20 years, which is 14.2339 using a 3.5% discount rate. The PV in this example is £142,339.
16.12

Annuity Factors may still be used where the constant stream of costs or benefits begins later than Year 1.

Example: What is the PV in the middle of Year 0 of 15 constant real annual costs of £75 starting in the middle of Year 7?

  1. First, calculate the PV as if the costs started in Year 1. Using the annuity factor for 15 years and a 3.5% per annum discount rate, the PV is £865 ( = £75 x 11.5281).
  2. However, given that the costs start in Year 7, this £865 figure represents the PV at the middle of Year 6.
  3. To convert this to the PV at the middle of Year 0, apply the discount factor for Year 6, which is 0.8135.
  4. The solution is thus £704 ( = £865 x 0.8135) or ( = £75 x 11.5281 x 0.8135).