Signature
NPV = (BCR - 1) * PVC
| Inputs | Definition | Unit |
|---|---|---|
BCR | Present value of monetised benefits divided by the present value of costs, as in HE-FM-CBA-002 | ratio |
PVC | Present value of the costs in the ratio's denominator | the same currency and price year as NPV, above zero |
NPV | Present value of monetised benefits minus the present value of costs, recovered from the ratio | currency in the stated price year, £ million in the examples |
|---|
Function
Benefit-cost ratio comparison and classification function
Maps the present values of monetised benefits and costs of one or more options, split where needed by how consequences are classified and by who bears the costs, to ratio summaries and to their link with net present value. The basic ratio and net present value are given under cost-benefit analysis (HE-FM-CBA-002 and HE-FM-CBA-001); PVB and PVC on this page are the same quantities as PV_B and PV_C there. The formulas below relate the ratio to net present value, apply it between mutually exclusive options, and show how the classification of savings, transfers and costs outside the public sector changes it.
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Implementations
Excel
Net present value from a ratio and its costs in one cell
Excel subtracts one from the ratio and multiplies the result by present-value costs, using named cells BenefitCostRatio and PVCosts.
=(BenefitCostRatio-1)*PVCosts
Assumptions
Positive costs in the denominator of a reported ratio
PVC is greater than zero. With zero or negative costs the ratio is undefined or changes sign, and net present value has to be computed directly from benefits and costs.
Ratio and costs taken from the same classification
BCR and PVC come from the same analysis, with the same treatment of savings and transfers. Net present value does not depend on that treatment, but a ratio and a cost taken from differently classified versions of an analysis return the wrong figure.
Worked examples
Local pilot with a benefit-cost ratio of 3
In the article's illustration, a local pilot with a ratio of 3 and costs of £10,000 has a net present value of £20,000.
BCR = 3; PVC = 10000; NPV = 20000
National programme with a benefit-cost ratio of 3
The same ratio on costs of £100 million gives a net present value of £200 million, ten thousand times the pilot's net value. The ratio alone does not show this difference in scale.
BCR = 3; PVC = 100000000; NPV = 200000000
Routine vaccination option X recovered from its gross ratio
Option X in the article's vaccination example has a gross ratio of 2.0 on programme costs of £4.0 million, so its net present value is £4.0 million, matching benefits of £8.0 million less costs of £4.0 million.
BCR = 2.0; PVC = 4.0; NPV = 4.0
Common errors
Reading a reported return on investment as a benefit-cost ratio
A return on investment is the ratio minus one, but some studies label a benefit-cost ratio as a return on investment. A reported return of 4 to 1 can therefore mean a ratio of 4 or a ratio of 5; on costs of £1 million the implied net present value is £3 million or £4 million.
Ranking options by ratio without their costs
Two options with the same ratio can differ in net present value by any factor: the pilot and the national programme above both have a ratio of 3, with net present values of £20,000 and £200 million. Ratios reported without costs cannot rank options by the value they create.
Sources
Masters and colleagues on return on investment as the ratio minus one
Masters R, Anwar E, Collins B, Cookson R, Capewell S. Return on investment of public health interventions: a systematic review. Journal of Epidemiology and Community Health. 2017;71(8):827-834. Introduction, which defines the cost-benefit ratio as benefit divided by cost and the return on investment as that ratio minus one.
Turner and colleagues on returns on investment reported as ratios
Turner HC, Hori Y, Revill P, Rattanavipapong W, Arai K, Nonvignon J, Jit M, Teerawattananon Y. Analyses of the return on investment of public health interventions: a scoping review and recommendations for future studies. BMJ Global Health. 2023;8(8):e012798. Introduction, which gives the standard formula for return on investment as benefits less cost divided by cost, and the results, which note studies whose reported returns appeared to be benefit-cost ratios.
Canonical Identity
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