# Definitions & Mathematical Basis of Benefit/Cost

Benefit/cost analysis is a tool for analyzing the relation between the costs and the benefits of a project, and comparing similar projects on the basis of those results. These two objectives, relating and comparing, determine the structure of the tool. Benefit/cost analysis is a ratio analysis; the relationship between benefit and cost is directly expressed as the quotient of one divided by the other. The relationship might be analyzed by the net of one from the other (i.e., subtract costs from benefits), but such a procedure would not take account of the scale of the project. Is a net benefit of \$5000 significant? It depends on whether the total costs are \$1000 or \$1,000,000. A ratio calculation takes into account not only the magnitude of a difference, but also its scale in relation to the original terms.

There is a second advantage to a ratio analysis: It is easily expressed in a single value (the result of dividing out the ratio). This quotient can then be compared across projects. In order to make this comparison, the projects must be roughly comparable in the first place. If they are not, if "costs" or "benefits" are defined differently for the projects, then a comparison would result in non-sense. The analysis could be mathematically correct but contextually gibberish.

There are two basic models for benefit/cost analysis. Basic benefit/cost analysis is concerned with total benefits and total costs. Add up all the benefits from a project and divide them by the total costs. The result is an analysis of the total effect of the project. There is a second model, derived from the first, called "effectiveness/cost analysis." In this model, the benefit per unit is divided by the cost per unit. It is used when a project provides multiple modules of basically the same service or product. The emphasis is on efficiency, rather than total effectiveness.

Benefit/cost analysis and effectiveness/cost analysis will result in the same conclusion when a project is made up of identical modules which consume the total budget with no remainder. The analyses will return different conclusions if a project includes modules with different efficiency ratios, or if there are costs incurred for benefits which are not fully used (e.g., project administration staff who could supervise another three modules, were there enough money to fund them).

Either model may include discounting in its design. The mathematics of discounting is straightforward, if a little tedious. The discounted value of a future benefit (or cost) is the value of the benefit divided by the discount rate for that future year. The discount rate is

(1 + r)

where "1" represents the value in the present and "r" represents the rate at which that value changes. The value of the discount rate for any period in the future is

(1 + r)n

where "n" is the number of years in the future to which you wish to project. The total process for discounting over time is

PV = SUM(b/[1+r]n)

Or, the present value of a stream of benefits is the sum of the discounted benefits for each year. The discounted benefits are the future value of the benefit divided by its discount rate for the specified future year.