BC-4 

August 12, 1994 
(TECBU2) 

TECHNICAL BULLETIN


TO: All Benefit-Cost Analysts 
FROM: Gary Sepulvado, Cliff Oliver, Ken Goettel, and Jerry Horner 
SUBJECT: The Mathematics of Aggregating Benefit-Cost Ratios 

This fulfills our promise to commit to paper the procedures of aggregating benefit-cost 
ratios (BCRs) when a large number of residential structures are under consideration in 
flood hazard areas. In several communities flooded during the Midwest Flood of 1993, 
for example, the number of structures coupled with the urgency of resolving BCR 
questions overwhelmed the expertise of the few staff analyst that were available at the 
time. It was therefore appropriate to ask the question whether there is an acceptable 
mathematical solution that will produce results expeditiously and accurately. This 
bulletin shows how to calculate the BCR of projects, groups and subgroups of residential 
structures expeditiously and accurately. 

Two things are considered in this bulletin: (1) how benefit-cost results are correctly 
aggregated, and (2) when it makes sense to use benefit-cost results that are aggregated. 
The following examples demonstrate the procedure to be used in aggregating benefit-cost 
results. 

1. Aggregating Benefit-Cost Results 
When confronted with a group of residential structures at different elevations, they may 
be grouped by elevation “bands” or contours, and for each group at the same elevation a 
benefit-cost analysis may be performed. 
(This assumes all other things about the structures are fairly equal as to vulnerability, 
such as building type, contents, age, etc.) 

The proper way of calculating an aggregate BCR for any such group of structures is to 
add up all the costs and all the benefits, thus producing the total costs and total benefits 
for the structures involved in any one group, and then calculate the BCR by dividing the 
total benefit by the total cost. Note that it would be mathematically wrong to average the 
BCR of any two structures, groups, and subgroups however. Directly averaging BCRs 
gives erroneous results. Therefore, BCRs must never be averaged. 

Example of Aggregating Benefits and Costs 
This is a table with four columns and four rows.
Column one is the project, column two is the cost, column three are the benefits, and column four is the BC ratio.
The first project in column one is Group A with a cost of $10,000, benefits of $25,000, and a BC ratio of 2.5. 
The second project in column one is Group B with a cost of $1,000,000, benefits of $500,000, and a BC ratio of 0.5. 
The third project is the Aggregate with a cost of $1,010,000 (A + B), benefits of $525,000 (A + B), and a BC ratio of 0.52 (Column 3 divided by 2). 

Note: “Averaging” the BC ratios in the example above gives wrong results (2.5 + 0.5 
divided by 2 = 1.5, compared to the correct BCR of 0.52. Averaging BCRs does not 
recognize the weighted difference among projects. In this example, the Group B project 
has 100 times the weight of the Group A project (one million vs. ten thousand). 
Therefore, averaging would produce a false BCR of 1.5, in this example. 

2. When Aggregating Benefit-Cost Results Makes Sense 
Aggregating benefit-cost results makes sense if and only if the projects and the groups 
within them are sensibly related. For example, when a whole neighborhood or 
subdivision is being considered for a “buyout,” it makes sense to aggregate benefit-cost 
ratios. Another example is buying out structures with benefit-cost ratios of less than one 
to avoid a pattern of “checkerboarding.” Aggregation to prevent checkerboarding makes 
sense because it facilitates the conversion of land areas to open space by removing 
structures where a majority of structures have BCRs greater than one. If an aggregate of 
the whole project has a BCR greater than one, then it meets hazard mitigation regulatory 
requirements under the Stafford Act. 
Keep in mind, however, that “averaging” several distinct projects does not make sense. 
Also, a project or group in one neighborhood or city cannot be combined logically with a 
project in another neighborhood or city to demonstrate that the combined project has a 
BCR greater than one. Each project must have a BCR greater than one to satisfy 
regulatory and statutory requirements under the Stafford Act.