Allowing for Uncertainty

Abstract

The techniques of project analysis have been considered so far as if the basic data, which they use, is known with certainty. However, both technical and economic information is used in the form of forecasts, and is subject to considerable uncertainty. It is possible to conceive of different values, based on experience, for the fundamental technical relations in any productive process, and for the project costs and benefits at either financial or economic prices. For most project data, a range of values can be found or predicted, yielding the possibility of different conclusions in the application of project worth measures.

Appendix: Option Value and the Value of Waiting

The additional information to be obtained by waiting for committing to a final project decision offers another way of addressing uncertainty. The discussion of an option value, based on the value of waiting, occurs in both the financial and environmental economics literature and is based on the premise that

• 1. an investment decision on a project is generally irreversible, at least in the short-term, and rules out other uses of the resources involved

• 2. when the future is uncertain, delaying a project will provide more information on benefits and costs in later years.

The option value is the difference between the net benefits from a project (usually expressed as an NPV) after a delay and the net benefits if the project is undertaken immediately. Where this is positive, it gives the benefit of waiting. By definition, a positive option value implies an immediate decision to invest is not correct and will not maximise net benefits. Ignoring a positive option value imposes a cost as it means future benefits are being lost. Hence the argument that if a project goes ahead immediately, the positive option value should be added to project investment costs.

A negative option value arises where the costs created by delaying benefits exceed any additional information obtained by waiting. In this case the decision to invest immediately will maximise net benefits.

A helpful explanation of the logic involved is given in the context of a project involving development of a forest area. The choice will be between ‘Conservation’ to preserve the ecological services offered by the forest or ‘Development’ to create marketable products, such as crops or timber. For simplicity, the discussion is for two periods. In period 0 (i.e. the present), the choice is Development, which rules out future conservation, and Conservation, which allows the option in period 1 of either developing or conserving. Development benefits are D₀ and D₁, for the two time periods are assumed to be certain. The conversion benefits in the present (V₀) are known with certainty, but in period 1, may be either high (V_high) with a probability of p or low (V_low), with a probability of p − 1.⁵

A conventional cost-benefit comparison, which ignores the option of waiting, compares Conservation and Development. Hence Development is justified if its expected net benefits (ED) exceed those of Conservation (EP).

Hence to justify Development requires

$$ {\mathrm{D}}_0+{\mathrm{D}}_1>{\mathrm{V}}_0+\mathrm{p}.{\mathrm{V}}_{\mathrm{high}}+1-\mathrm{p}.{\mathrm{V}}_{\mathrm{low}} $$

Waiting one period entails allowing the forest to function normally with a known benefit of V₀ and investing the funds that would have been spent on the project elsewhere for one period at the discount rate, so their present value remains constant. On the assumption that more information will be known on the uncertain Conservation benefits in year 1, these may be either pV_high or 1 − pV_low. If they are low, the Development option can be taken, but if they are high, the Conservation option may be justified. Hence, with waiting, there are three possible outcomes

• (a) waiting and then developing, so total benefits over the two periods (ED) are V₀ + D₁

• (b) waiting and then conserving with high conservation benefits, so benefits (EP) are Vo + pV_high

• (c) waiting and then conserving with low conservation benefits, so benefits (EP) are Vo + 1 − pV_low

The optimal decision will depend on the size of ED relative to EP, which in turn depends on the uncertain benefits from conservation. The probability p will not be known in the period 0 when the immediate decision on whether to invest has to be taken. By assumption in this example, it is a random value in period 0, that will be known in period 1, because waiting has provided more information. If V_low occurs, the decision should be to wait and develop, but if V_high occurs, the decision should be to wait and conserve.

The value of waiting (EW) in period 0 and then with more information choosing the best option in period 1, is given by the expression

$$ EW={\mathrm{V}}_0+{pV}_{\mathrm{high}}+\left(1-\mathrm{p}\right){\mathrm{D}}_1 $$

The logic is that waiting and not developing gives a known conservation value in year 0 of V₀. In year 1, if V_high occurs, conservation will be chosen, and if V_low occurs, the choice will be development with a benefit of D₁. Hence in year 1, benefits are a probability-weighted sum of V_high and D₁. The economic value of waiting EW will always be higher than the immediate choice of conservation (EP) provided D₁ is greater than Vlow and p is less than 1.0

The second comparison must be with the choice of immediate development ED. Immediate development is assumed to give certain benefits so

$$ ED={\mathrm{D}}_0+{\mathrm{D}}_1 $$

Hence ignoring the option of waiting requires that immediate Development creates benefits greater than waiting, so ED > EW. Alternatively, where waiting has more benefits, so EW > ED, option value is the loss EW − ED created by taking an immediate decision to develop.⁶ The argument is that the decision rule to justify Development should be the requirement ED > EW, not the weaker one that benefits of immediate development exceed those of continuing with conservation (EP), so ED > EP. Where the value of waiting (EW) exceeds benefits from both immediate development (ED) and continued conservation (EP), the difference between EW and the larger of ED and EP gives the negative option value created by taking an immediate decision. In theory, it can be seen as a cost imposed by foregoing the option of waiting and should added to the cost of a project, where waiting is ignored.⁷

In practice, it is rare to see detailed option value calculations included as part of project costing; however, the general principle that decisions can be delayed to allow more information to be collected is an important one. This is particularly the case where projects involve relatively large irreversible investments. Where waiting allows more information over the future (e.g. in the growth of demand) it provides a means of addressing uncertainty over the probability distribution of future outcomes. As discussed in Chap. 3, conventional analyses of project timing, to compare alternatives with different start dates by the NPV (discounted to the same base year), may forgo this opportunity, as it makes no allowance for the availability of additional information.