Roberto P. Lima Netto
The benefit-cost (B/C) analysis is a method used to study investment proposals. Although not a new method, its widespread and accelerating popularity is a recent development because governments and development institutions, as well as private investors, are today requiring much more sophisticated analysis of proposals before committing funds.
In general terms, the application of the B/C method involves the following steps:
The quantification of the benefits and costs involved in the proposal and its alternatives;
The discounting of the benefits and costs in order to obtain one figure to represent the value of all the benefits and another to represent the value of all the costs at the present time;
The use of these figures to compute B/C ratios to choose among the alternatives.
Quantification of Benefits and Costs
Each investment proposal must have its benefits, or it would not warrant study at all. If we are studying a proposal to construct a flood-regulating dam, we look for advantages of the investment such as the reduction in damaged crops. But the problem is not so simple. We must make an evaluation, that is, put a figure on these benefits. Using the available information, a monetary figure must be assigned for the future years following the construction of this dam.
On the cost side we have, of course, not only the investments necessary for the construction but also a yearly cost for maintenance of the dam.
It is obvious that the assigning of monetary values to benefits and costs is a difficult and time-consuming job. But that is not a handicap of the B/C method because any other rational method of investment analysis also has to pass through this step.
The example of the dam given here presents the problem in its most elementary form. Let us suppose that the construction of the dam will create a lake that will make the region a resort area and will lead to the construction of motels, restaurants, facilities for water sports, and so on. The new lake will bring more economic activity to the region, and, if we want to take this benefit into consideration, we have to put a monetary figure to measure this advantage to the community. Other benefits and costs might well be involved, as for instance, environmental costs very much in evidence for some industrial projects.1
Discounted Benefit and Cost
The first step, then, is to obtain monetary figures for yearly benefits and costs. They may be—and generally are—different from year to year. What we want here is a number that represents, at the present time, the value of the present and future benefits and another number to represent the value of present and future costs.
The concept is simple. Let us suppose that you have part of your savings in a savings account paying 5 per cent a year compounded at the end of the year. Therefore, each $100 that you have now is worth $105 a year from now. A member of your family applies to you for a loan of $100 to be paid after two years. Since it is a family matter, you do not want to make any money in the transaction, but you do not want to lose money either. You know that if you leave your money in the savings account you will get $105 after one year, and, if you leave this money in the account, $110.25 after two years. You forego $100 now to get $110.25 in two years; therefore, the present value for you of $110.25 in two years is at least $100. The minimum you should receive after two years is $110.25 to pay for the loan of $100.
Back to the benefits and costs. Given a certain interest rate you can compute the present value of the benefits and this we will call benefit (B) of the project. You can also compute the present value of the costs that we will call cost (C) of the project. These computations are very easily made with the help of financial tables.2
Having done these computations, we now have one figure to measure the present value of all the benefits and one figure to measure the present value of all the costs of the proposal, and we are through the second step. (The question of which interest rate to use is complex and will be dealt with later in this article.)
Criterion for Proposal Acceptance
Is the project acceptable or not? Let us assume, to begin with, that there is just one proposal and that the problem of choosing among several alternatives does not arise. The B/C of this project is defined as the ratio of the benefit to the cost. If this ratio is greater than one, we accept the proposal; such a ratio means simply that the benefit of the proposal is greater than the cost. If the benefit-cost ratio is less than one, we reject the proposal. If the ratio is equal to one, we should be indifferent toward accepting or rejecting the proposal. The following example might clarify this point: The construction of a dam is proposed at a certain site. The cost of building this dam is 1,000 monetary units. If the dam is not built, we can expect annual flood damage in the region equal to 300 monetary units. With the project, these damages will be reduced to 70 monetary units. In addition, the annual cost of maintenance of this project is 50 monetary units during its ten years useful life. After this period a bigger hydroelectric project is planned to be completed downstream and this dam will no longer be useful. Let us also assume a discount rate of 8 per cent.
The annual benefits of this project are the 230 monetary units of prevented flood damages less the 50 monetary units annual maintenance expense. Discounting this stream of funds at 8 per cent, we get a figure of 1,210 monetary units for the benefit of the project. The cost of the project is already a present-value figure since it occurs in the present year. Therefore, the cost of the project is 1,000 monetary units. The B/C ratio is 1.21 and because it is higher than one, the project is acceptable.
Let us suppose that you finished the above analysis and presented it to the responsible authority. There may be a disagreement about your B/C ratio. The figure that the authority has calculated is only 1.16 as opposed to your 1.21.
After some talk, you find out that they considered the 50 monetary units of annual maintenance expenses on the cost side instead of as an offsetting factor of the benefits as you did. If we do that, the benefit of the project will now be 1,546 monetary units, the present value of (300—70) monetary units every year for ten years discounted at 8 per cent. The cost of the project will be 1,336 monetary units, the present value, at 8 per cent, of the initial 1,000 monetary units and the annual maintenance cost. The new B/C ratio will be 1.16, although the difference between benefit and cost is the same—210 monetary units—in both cases. Which is right? The answer to this question is that they are both right; there may be more than one appropriate B/C ratio.
But, if the B/C ratio is not unique, could we not have an instance where it varies from below unity to above unity, when a decision to accept or not to accept the proposal would be subject to the allocation of items as cost or negative benefits? The answer is no. It can be proved that if a project has a B/C ratio greater than one, all its possible B/C ratios will be greater than one. Similarly, if the proposal has one B/C ratio smaller than one, all the possible B/C ratios will be smaller than one. This is easily proved because the only result of this manipulation is to increase (or decrease) the benefit while increasing (or decreasing) the cost by the same amount, keeping the difference between benefit and cost constant, as seen in the example above. Therefore, if the benefit is greater (smaller) than the cost and consequently the ratio is greater (smaller) than one, any manipulation that increases or reduces both benefit and costs by the same amount will not change the situation. The ratio may change but if it is greater (smaller) than the unit, it will remain in the same way after the change. Therefore, although the B/C ratio is not unique, the acceptance criterion is valid.
The Choice Among Alternatives
The B/C method would be very straightforward if only one proposal were being studied. That is not what actually happens where several alternatives are available to attain the same objective.
Using the same flood control problem, let us suppose that alternatives exist. We can either build a dam in location A or in location B. Dam A is the one presented above, whose B/C ratio, we just computed. Dam B would involve an initial investment of 500 monetary units, bring about a net annual flood damage saving of 200 monetary units, and have an annual maintenance of 100 monetary units. Depending on whether we consider the maintenance expense as offsetting the benefits, or as a cost, we have a benefit cost of 1.34 or 1.15, respectively.
The problem now is to define a criterion to choose between the two alternatives. The first one that occurs to us is to choose the alternative with the highest B/C ratio. But as the B/C ratio is not unique, which one should we consider? In the above example, we get conflicting decisions depending on the choice of the method to compute the B/C ratios we make, because we have the following ratios:
|Method 1||Method 2|
Therefore, the criterion based on the highest B/C ratio is not good.
The correct criterion to choose among mutually exclusive alternatives using the B/C method is to use an incremental analysis. Let us assume that we have four mutually exclusive alternatives each lasting ten years, having the following present value of benefits and costs (Table 1).
We first rank the alternatives by increasing cost, as is already done in the table above. After that, we eliminate all the alternatives having a ratio below unity. In the specific case discussed above, all the alternatives have B/C ratios larger than one, and, therefore, no one can be deleted in the outset. Then we compute the incremental B/C ratio, as shown in Table 2.
We delete every project having an incremental ratio smaller than one and we compare the next alternative with the last one not deleted. We choose the last alternative having an incremental B/C ratio greater than unity. So we compare alternative B with A, which precedes B, and find an incremental B/C ratio larger than (2.25). Therefore, we delete A in favor of B. We next compare C with B. Notice that we would compare C with A if the incremental B/C ratio for B and A were smaller than one, because if this were the case, we would have deleted B. The incremental B/C ratio between C and B is also larger than one (1.50) and we delete B in favor of C. Finally, we compare D and. C. Since the incremental B/C ratio is smaller than one (0.50), we delete D in favor of C. C is our best alternative.
The rationale for this method is quite simple. Initially, we drop all proposals with a B/C ratio smaller than one because they are clearly not worthwhile. Then, when we compare one alternative with the immediately less costly one not yet deleted and the incremental B/C ratio is smaller than one, we consider that incremental investment also not worthwhile. We stick with the less costly project and drop the other.
The Discount Rate to Be Used
As indicated earlier, one of the major complexities in B/C analysis concerns the interest rate to be used in discounting the benefits and costs involved in the proposal.
For example, take an individual’s investment opportunities. He has his money invested at interest rates equal to or above 7 per cent a year. In his case, he will appraise any investment proposal using a discount rate of 7 per cent, all risks being assumed equal. The reason for this is that 7 per cent is the minimum he is getting for his money. If he uses the 7 per cent as the discount rate for benefits and costs of the proposal and the proposal is acceptable, this means that it yields more than 7 per cent and therefore he should allocate a part of his money invested at 7 per cent to this more attractive proposal. This 7 per cent is the “opportunity” cost of his money.
For firms, the discount rate that may be used is also the opportunity cost of capital. The determination of this discount rate is a difficult problem; moreover, the use of this discount rate carries the assumption that there is no capital rationing—that is, that money for all purposes can be borrowed at the same rate of interest that applies to the project we are considering. Several firms define a cutoff rate for the proposals, and this should be used as the discount rate.
For government agencies it is quite common to use the cost of borrowing money as the discount rate. For example, let us assume that a regional government issues bonds at a cost of 7 per cent to raise money for the construction of a dam. We may use this 7 per cent as the discount rate. One of the assumptions made is that there is no capital rationing and the regional government is able to raise the amount of money necessary to finance all other regional projects at the same rate.
Other methods of analyzing investment proposals are also valuable in the appropriate circumstances. There need be no quarrel between them, since, if consistently applied, they should all provide the same results as the straightforward B/C method.
For a discussion, see Michael L. Hoffman, “Development Finance and the Environment,” Finance and Development, September 1970.
For a detailed explanation, see George B. Baldwin, “Discounted Cash Flow,” Finance and Development, September 1969.