Discounting Benefits and Costs Over Time
4. ALTERNATIVE BENEFIT-COST DECISION CRITERIA
Once the analyst has measured the benefits and costs over the life of the project for each affected resource output, the next step is to summarize this information in a bottom line. For the information to be most useful for decision making, it is helpful to compare the sum of the discounted benefits produced by the alternative to the discounted costs incurred by the alternative.
The criteria and summary measures which are most defensible and commonly used include: (1) Net Present Value, (2) Benefit Cost Ratio, and (3) Internal Rate of Return. We will discuss each of these individually and
then discuss under what conditions they yield exactly the same ranking of alternative policies or projects.
4.1 Net Present Value (NPV)
As the name implies, Net Present Value (NPV) is the present value of the benefits minus the present value of the costs. The difference is the net gain adjusted for the timing of benefits and costs. The units of measurement are present worth of dollars in the base year in which all of the benefits and costs are figured.
The definition of NPV can be understood by reference to equation 8-2:
As equation 8-2 illustrates, the benefits in each year are subtracted from the costs in each year, and this difference is discounted back to the present time period (t=0). If the resource management action's present value of benefits exceed the present value of costs, then NPV will be greater than zero, i.e., positive. In general, the larger the NPV is, the more net benefits society realizes from the resource management action.
If the analyst is using NPV as a decision rule, then only projects with NPV greater than zero are accepted as economically efficient. Projects with negative NPVs mean society gives up more than it gets over the life of the resource management action. NPV is a discounted measure of whether the resource management action represents a Potential Pareto Improvement over the life of the project. Taking benefits as measures of gains and costs as a measure of losses, if NPV is greater than zero, then the gainers over the life of the project could compensate the losers over the life of the project and still have some benefit remaining.
4.2 Benefit-Cost Ratio (BCR)
The Benefit Cost Ratio (BCR) takes the same information used in equation 6-2 but expresses it as a ratio. Specifically, BCR equals the present value of the benefits divided by the present value of the costs.
Equation 8-3 presents this relationship:
Since BCR is by definition a ratio, it will always be a positive number.
If the discounted benefits exceed the costs, then BCR will be greater than one. Thus a BCR of 3:1 means that society receives three dollars worth of benefits for each dollar of cost. Thus BCR is an average return yielded by the project or resource management action.
While it is sometimes believed that a higher BCR is better, a very high benefit cost ratio may mean the project is too small, and its size and scope should be expanded. If BCR is being used as a decision rule, then only projects or resource management actions that have BCR greater than one would be considered for adoption.
4.3 Internal Rate of Return (IRR)
The Internal Rate of Return (IRR) is the percentage return on money invested in the project that equates the present value of the benefits with the present value of the costs. In essence, IRR is the earning power or interest earned when the money is invested in the project under study.
Mathematically, IRR is that interest rate which results in the NPV of the project equaling zero. Thus the IRR formula solves for r in Equation 8-4 such that NPV = 0 or:
Unlike the BCR or NPV, which can easily be computed on even a simple calculator, determining the IRR involves a series of relatively complex calculations. As such it is best performed using financial calculators that are programmed to calculate IRR, or a computer software
program designed to calculate IRR. By itself, this added computational burden is not a valid reason for reluctance to use IRR (but there may be other objections that will be explained below).
To use the calculated IRR to determine the economic efficiency of a project involves a simple comparison. The calculated IRR is compared to the interest rate at which the agency or society can borrow money. As will be recalled from our earlier discussion, one view of the interest rate is the opportunity cost of foregone rates of return in society's next best investment. Thus the IRR decision rule is to recommend any project which generates an Internal Rate of Return greater than the interest rate.
4.4 Does It Matter Which Discounted Measure Is Used?
A comparison of the three formulas for NPV, BCR and IRR indicates that all three use the same information. Therefore, one would expect these three measures would yield the same conclusions on which projects should be adopted and which should be rejected. However, this is only true under certain conditions.
The conditions under which all three measures will yield the same conclusions regarding the desirability of projects are: (1) Either there are no budget limits that preclude adopting all economically efficient projects that pass their respective decision rule OR the size of projects in terms of investment costs are similar. (2) The projects or policies are not mutually exclusive. That is, each project could be implemented without precluding implementation of another. (3) Either the projects being compared have a similar length of time (i.e., economic life) OR the reinvestment opportunities yield the same returns as the original project.
If these conditions are not met, what appears to be the best project alternative will vary depending on which discounted measure (NPV, BCR, IRR) is used to evaluate the alternatives. For example, Table 8-2 involves violation of both aspects of condition 1 and part of condition 3. First, the projects have substantially different investment requirements. If we add to this the budget constraint that society only has $10 million to invest, then we must choose between the projects rather than adopting both (which is warranted since both are economically efficient regardless of whether one uses NPV or BCR). In addition, we are unable to find reinvestment opportunities that would yield the same return as we get from doing several project As (thus the default assumption is that the remaining $8.3 million of the original $10 million not invested under alternative A would earn just the discount rate).
Under these circumstances, which project is best to select depends on whether we rank by NPV (choose Project B) versus rank by BCR (choose Project A).
An example of this type of situation is the decision whether to build a light rail or to add more buses. The light rail will have much higher investment costs than adding more buses, yielding a lower BCR, but higher NPV. Since we are likely to find situations where governments have very limited budgets to devote to public projects, or nature makes the projects or policies mutually exclusive, it is important to know the advantages of each of the three discounted measures of project worth.
4.5 Advantages of NPV as a Theoretically Correct Measure of Net Benefits
We can illustrate the merits of NPV relative to BCR by reference to Table 8-2. When conditions 1-3 above do not hold, the choice between Plan A and Plan B becomes clear: does society prefer $6 million in net benefits or $2.3 million in net benefits? If all of the opportunity costs are correctly measured in the present value of costs, and the discount rate accurately reflects the foregone investment opportunities, then Plan B is preferred. People prefer more ($6 million) to less ($2.3 million) if these are mutually exclusive projects and we cannot invest in several identical project As. Our theory of what is an improvement in well-being is written in terms of maximizing the total amount of the net gain, not in terms of maximizing the rate of return. A very high rate of return for which we are restricted to collecting on only a $1 investment is not attractive if it means giving up a larger total return on some other investment. (Note that if the remaining
$8.3 million not invested in project A just earns the discount rate, then this would pull down the BCR of project A to 1.23, again demonstrating the superiority of project B).
Another way to see that project B is superior to project A is to compare the marginal benefits and the marginal costs of the additional expense involved in B. While Project B includes $8.3 million more in costs, it provides $12 million more in benefits, justifying the additional expenditure.
As Boadway (1979) has shown, maximizing NPV is a necessary condition for a project or policy to result in a Potential Pareto Improvement.
Maximizing the BCR is not strictly related to attainment of Potential Pareto Improvement, since the BCR is the average rate of return. A high average rate of return tells us little about whether the total benefits are maximized.
A high average return can be consistent with a project which is not large enough (marginal benefit from expansion greater than marginal cost of expansion, i.e., MB > MC). To attain the maximum gain, marginal cost must equal the marginal benefit (MB = MC). Thus NPV is a useful criterion for determining the optimal scale of a resource management action as well as timing of implementation. The BCR is an ambiguous indicator of the timing or the scale.
Additionally, the actual value of the BCR can be influenced by how costs and benefits are defined. For instance, suppose that a project with costs of $5 million will increase developed recreation by a value of $20 million but reduce dispersed recreation by $10 million. Is the reduction in dispersed recreation a reduction in benefits, or an increase in costs? In the former case, the BCR will be ($20 - $10 million)/($5 million) = 2; in the latter case, the BCR will be ($20 million)/($10 + 5 million) = 1.33. In either case, the NPV = $5 million. The BCR will always be greater than 1 if the NPV is positive (and less than 1 if the NPV is negative), but the exact value of the ratio depends on whether foregone benefits are treated as a cost and placed in the denominator or treated as a negative benefit in the numerator.
4.6 Advantages of BCR
There are, however, a few merits to use of the BCR. Some authors claim (Sassone and Schaffer, 1978:21) that, if one must rank many projects within a budget constraint, use of BCR and calculating incremental BCRs is the appropriate procedure. In addition, the BCR is easily understood by the public because it is expressed as dollars of benefit per dollar of cost. Thus it has a nice intuitive appeal. Lastly, in comparing projects where one does not want the scale of the project to influence opinions about desirability, the BCR is unaffected by the scale of the projects.
4.7 Merits of IRR as a Measure of Discounted Net Benefits
So far we have focused on the relative merits of NPV and BCR.
However, the IRR has a similar set of advantages and disadvantages.
Because the IRR can be expressed as a return per dollar of investment, it shares with the BCR its ease of understanding by the public and being unaffected by the scale of the project. However, since it also shares BCR's drawback of using a rate of return rather than total return as the decision
rule, it too suffers from being a poor indicator of which project provides the greatest total gain to society. What is worse about IRR is the possibility that there exist several interest rates which set NPV = 0. Depending on the pattern of costs and benefits over time, there may be two or three or more interest rates (r) in equation 8-4 that are possible answers. Unfortunately, the analyst is likely to be completely unaware of exactly which interest rate has been solved for.
However, the IRR has one advantage not shared by BCR and NPV: the ability to make comparisons between alternative projects or policies without first having to specify an exact discount rate. In essence, the IRR formula solves for the discount rate (i.e., IRR) which would just make the present value of benefits equal to the present value of costs. If the actual discount or interest rate is lower than the internal rate of return, then this project will provide net benefits in excess of its opportunity cost. Hence this project or policy option would be acceptable on economic efficiency grounds.
However, in many instances an agency or decision-maker need not state the exact discount rate it is using. As long as the project or policy generates a
"reasonable" return (given the implicit discount rate the agency is using),
the project can be accepted without ever having to decide the exact discount rate to use. The practical advantage of such an approach should be obvious given the earlier discussion about the difficulty in getting universal agreement on what should be the appropriate discount rate.
However, as also discussed earlier, most agencies have their discount rate specified, so this feature of IRR is not always compelling. In addition, many agencies have their decision criteria (i.e., NPV, BCR and IRR) specified as well. Agencies such as the Forest Service, Army Corps of Engineers, and USDA Natural Resources Conservation Service generally rely on use of NPV, although BCRs are also computed as supplemental information.
In general, use of NPV as the Benefit-Cost Analysis decision criteria has a great deal to offer. It is the most theoretically correct measure in the sense of being most consistent with the conceptual foundation of BCA. NPV does not suffer the multiple solutions problem that can potentially plague IRR calculations.