Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Benefit-cost ratio comparison and classification function

r(PVB, PVC) = (NPV, BCR_YX, BCR_gross, BCR_net, BCR_T, RPSC)

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.

  • Net present value recovered from a benefit-cost ratio

    NPV = (BCR - 1) * PVC

    Recovers net present value from a benefit-cost ratio and the present value of costs it was built on. It rearranges the identity that the ratio minus one equals net present value per unit of cost, so a ratio above one and a positive net present value are the same finding whenever costs are positive. The ratio minus one is also the return on investment as defined in public health reviews, so a reported return converts to a ratio by adding one.

  • Incremental benefit-cost ratio between two mutually exclusive options

    BCR_YX = (PVB_Y - PVB_X) / (PVC_Y - PVC_X)

    Divides the extra present-value benefits of option Y over option X by its extra present-value costs. When Y costs more than X, an incremental ratio above one is the same condition as Y having the higher net present value, so the ratio applied incrementally gives the same choice as net present value. Average ratios of each option against doing nothing do not.

  • Benefit-cost ratio with averted costs counted gross or netted

    BCR_gross = (H + S) / C; BCR_net = H / (C - S)

    Gives the two ratios that arise when averted treatment costs are counted as a benefit (gross) or subtracted from programme costs (netted). Net present value, H plus S minus C, is the same under both, but the ratios differ, and the netted ratio is undefined or negative when averted costs reach or exceed programme costs.

  • Benefit-cost ratio with a transfer counted on both sides

    BCR_T = (PVB + T) / (PVC + T)

    Gives the ratio when a transfer, such as an incentive payment, is counted as a cost to the payer and an equal benefit to the recipient. The transfer cancels in net present value but moves the ratio towards one: a ratio above one falls and a ratio below one rises.

  • Return on public sector cost from benefits, non-public and public costs

    RPSC = (PVB - PVC_nonpublic) / PVC_public

    Divides present-value benefits net of costs borne outside the public sector by present-value public sector costs, as set out in the Green Book (2026). Costs that fall on patients, families or other sectors move from the denominator to the numerator, so the ratio measures social value per pound of public spending. It suits ranking options when the public sector budget is the binding constraint.