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Public Sector Discount Rates: A Comparison of Alternative Approaches

2 Discounting and Present Values

2.1  The need to discount

When appraising public projects a decision-maker is typically faced with decisions that deliver different profiles of costs and benefits over time. In such cases, the decision-maker typically cares not just about the absolute value of the costs and benefits, but also about when those costs and benefits materialise. For instance, a project that delivers a series of benefits in the near term is not usually considered to be the same as a project which has exactly the same pattern of costs, but delivers its benefits ten years later. Benefits that materialise sooner are usually considered to be more valuable than those that occur later.

How should alternative cost-benefit profiles be compared? One way of answering this question is to ask, 'What is the maximum amount that a decision-maker would pay now to secure the stream of net benefits available from a given project?' In other words, what is the present value of a given stream of net benefits? Discounting provides a way of answering this question by formally specifying the value that a decision-maker assigns now to outcomes that occur in the future. In doing so, discounting allows projects with different net benefit time profiles to be converted into single values with a common valuation date. These values can then be compared in order to reject proposals which do not yield a positive net benefit, and rank those that do.[3]

2.2  The Present Value of a Benefit Stream

It is therefore necessary to consider how the present valuation of any future cost or benefit depends on its timing. For example, how would the present valuation of $100 received T years in the future vary with T? As mentioned above, benefits that materialise sooner are usually considered to be more valuable than those that occur later (for reasons explored shortly).

At its simplest level, discounting amounts to little more than scaling-down the value of costs and benefits that occur in the future by a factor that increases with the length of the delay. The standard way of doing this is to scale the value of any cost or benefit down by a constant factor for each additional year into the future it is expected to occur. This constant factor is determined by the discount rate, and this approach is known as constant exponential discounting.

Suppose $1 is invested, earning an interest rate of r per period. After one period it is therefore worth $(1+r). Hence it is only necessary to invest $1/(1+r) in order to receive $1 after one period. The present value of $1 to be received in one period's time is thus $1/(1+r). The same argument leads to the result that the present value of $1 to be received in two period's time is $1/(1+r)2 . Hence the present value, PV, of the net benefit stream, x0, x1,x2, x3 … xT , is thus:

This can be interpreted as the maximum amount that the decision-maker would be willing to pay now to secure the net benefit stream under consideration. In other words, resolving net benefit streams into their PVs can be used to compare and rank alternative cost/benefit profiles.

2.3  Discounting over long periods

For any given discount rate, r, there is thus a discount factor for any period, t, equal to 1/(1+r)t. This obviously declines as t increases. Figure 1 demonstrates how discount factors decline, under constant exponential discounting for a range of discount rates, as the time period increases. The rate of decline is clearly much more rapid for higher discount rates. For example, under a 5% discount rate, $100m arising 50 years from now has a present value of roughly $9m.

A concern that has been raised with constant exponential discounting is that, even with a low discount rate, the weights that the decision-maker attaches to future time periods eventually become very small. In practical terms, this means that decision-maker effectively ‘stops caring' about outcomes in the future, provided these outcomes are distant enough.[4]

In response to this concern, some economists have proposed using time-varying or hyperbolic discount rates, where the discount rate is progressively reduced as the time when net benefits are received increases. This topic is considered in section 7.

Figure 1 Discount Factors for Alternative Discount Rates and Time Periods
Figure 1 Discount Factors for Alternative Discount Rates and Time Periods.

Notes

  • [3]An axiomatic approach to discounting was first proposed by Koopmans (1960); see Creedy and Guest (2008) for a simplified exposition. Without discounting, it is also not possible to compare infinitely lived alternative projects: however, on partial orderings using an ‘overtaking’ criterion, see von Weizacker (1965) and Heal (1998).
  • [4]It is convenient to think in terms of the ‘half life’, defined as the time taken for the value in that year to be reduced by half. Where r is the discount rate, the half life is equal to log2/log(1+r). For example, annual rates of interest of 2, 5 and 10% have half lives of respectively 35, 14.2 and 7.3 years.
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