Annualised capital charge net of discounted resale value

Signature

A = (1 - (1+r)^(-n)) / r; E = (K - S * (1+r)^(-n)) / A

Function

Capital outlay annualisation function at the evaluation discount rate

Computational function

• Computational function for annualised capital cost per unit of service from several capital inputs

  Takes the inputs a costing study usually holds, the price, resale value and useful life of each capital input, one discount rate and the annual volume of service, and returns the annualised capital cost per unit of service alongside the straight-line figure. It applies HE-FM-ANLS-001 to each input over its own life, sums the annual charges and divides by the volume. The Math / Code line shows two inputs, and the R and Python versions take any number.

  Inputs and outputs: K_1 and K_2 are the purchase prices of the two inputs, required, zero or above, in currency; S_1 and S_2 are their resale values at the end of their lives, zero if none, in currency; n_1 and n_2 are their useful lives or repeat intervals, required, in whole years; r is the annual discount rate, required, above zero, as a proportion; Q is the number of units of service a year, required, above zero; A_1 and A_2 are the annuity factors; E_1 and E_2 are the annualised charges and E_total their sum, in currency per year; c_unit is the annualised capital cost per unit of service and c_SL the straight-line capital cost per unit, in currency per unit.

  Assumption: Every input is annualised at the evaluation's discount rate with end-of-year sums, each input serves only the evaluated service, and the volume is the same in every year. A shared asset is first split between services as described on the Capital Cost page.

  Worked example (Retinal camera and screener training over 2,500 screens): The camera at GBP 50,000 with a GBP 5,000 resale value over 5 years and training at GBP 12,000 repeated every 4 years give annual charges of about GBP 10,142 and GBP 3,267 at 3.5%, or about GBP 5.36 per screen over 2,500 screens, against GBP 4.80 per screen on a straight-line basis, about 12% higher. K_1 = 50000; S_1 = 5000; n_1 = 5; K_2 = 12000; S_2 = 0; n_2 = 4; r = 0.035; Q = 2500; A_1 = 4.51505; A_2 = 3.67308; E_1 = 10141.66; E_2 = 3267.01; E_total = 13408.68; c_unit = 5.3635; c_SL = 4.80

  Worked example (Camera alone with no training cost): Setting the training price to zero leaves only the camera's charge of about GBP 10,142 a year, or about GBP 4.06 per screen, a limiting case that checks that each input is annualised separately. K_1 = 50000; S_1 = 5000; n_1 = 5; K_2 = 0; S_2 = 0; n_2 = 4; r = 0.035; Q = 2500; A_1 = 4.51505; A_2 = 3.67308; E_1 = 10141.66; E_2 = 0; E_total = 10141.66; c_unit = 4.0567; c_SL = 3.60

  Excel: =SUMPRODUCT((Price-Resale*(1+Rate)^-Life)/((1-(1+Rate)^-Life)/Rate))/Volume With one row per capital input in ranges named Price, Resale and Life, the rate in Rate and the annual volume in Volume, the formula returns the annualised capital cost per unit. =SUMPRODUCT((Price-Resale)/Life)/Volume gives the straight-line figure.

  R: capital_cost_per_unit <- function(K, S, n, r, Q) { E <- (K-S*(1+r)^(-n))/((1-(1+r)^(-n))/r); c(E_total = sum(E), c_unit = sum(E)/Q, c_SL = sum((K-S)/n)/Q) } Vectorised over K, S and n, with one element per capital input.

  Python: def capital_cost_per_unit(K, S, n, r, Q): E = [(k-s*(1+r)**(-m))/((1-(1+r)**(-m))/r) for k, s, m in zip(K, S, n)]; return sum(E), sum(E)/Q, sum((k-s)/m for k, s, m in zip(K, S, n))/Q Takes lists of prices, resale values and lives and returns the total annual charge, the cost per unit and the straight-line cost per unit.

  Test (Economic cost per unit exceeds the straight-line cost per unit): At a rate above zero the annualised charges sum to more than the straight-line charges for any inputs with a positive price. Expected result TRUE. Excel check =SUMPRODUCT((Price-Resale*(1+Rate)^-Life)/((1-(1+Rate)^-Life)/Rate))>SUMPRODUCT((Price-Resale)/Life)

  Test (Article's capital cost per screen reproduced): The camera and training inputs return GBP 5.36 per screen when rounded to the penny. Expected result TRUE. Excel check =ROUND(SUMPRODUCT(({50000;12000}-{5000;0}*1.035^-{5;4})/((1-1.035^-{5;4})/0.035))/2500,2)=5.36

  Common error (Annualising training over the equipment's life): Spreading the GBP 12,000 training cost over the camera's 5 years instead of its 4-year repeat interval gives about GBP 2,658 a year instead of GBP 3,267, and about GBP 5.12 per screen instead of GBP 5.36. Each input is annualised over its own life.

  Source: Tan-Torres Edejer T, Baltussen R, Adam T, Hutubessy R, Acharya A, Evans DB, Murray CJL, editors. Making choices in health. WHO guide to cost-effectiveness analysis. Geneva, World Health Organization, 2003. Section 3.3.2, which annualises each capital item over its own useful life, net of its discounted resale value, by dividing by the annuity factor A(n, r).

  A_1 = (1 - (1+r)^(-n_1)) / r; A_2 = (1 - (1+r)^(-n_2)) / r; E_1 = (K_1 - S_1 * (1+r)^(-n_1)) / A_1; E_2 = (K_2 - S_2 * (1+r)^(-n_2)) / A_2; E_total = E_1 + E_2; c_unit = E_total / Q; c_SL = ((K_1 - S_1) / n_1 + (K_2 - S_2) / n_2) / Q

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Implementations

• Excel

  Annualised capital charge net of resale value from named cells

  With the price in a cell named CapitalPrice, the resale value in ResaleValue, the rate in DiscountRate and the life in UsefulLife, the formula returns E. The built-in form =PMT(DiscountRate,UsefulLife,-CapitalPrice,ResaleValue) gives the same result.

  =(CapitalPrice-ResaleValue*(1+DiscountRate)^-UsefulLife)/((1-(1+DiscountRate)^-UsefulLife)/DiscountRate)

Assumptions

• Annualised sums falling at the end of each year of use

  Each annual sum is treated as falling at the end of its year. If the sums are treated as falling at the start of each year, the factor becomes 1 plus r, times A, and the annual charge is lower by the factor 1 plus r, so the convention has to match the timing of the model's other costs.

• Annualisation rate equal to the evaluation's real discount rate

  The rate r is the one used to discount the evaluation's other costs, so that capital and recurrent costs are valued on one basis, for example 3.5% a year in the NICE reference case (PMG36, section 4.5.1). It is a real rate, applied to prices held constant in one price year.

• Resale value counted whether or not the asset is sold

  S reflects the opportunity cost of the asset at the end of its useful life and does not depend on an actual sale, in line with the Green Book (2026) treatment of residual values.

• Replacement price for capital bought in earlier years

  For an asset already in use, K is its current replacement price and n a full useful life. Capital bought in earlier years can still be resold, so it still has an opportunity cost and is not treated as costless.

• Same service in every year of the useful life

  The asset gives the same service in each year of its life. Equipment that becomes obsolete before it wears out has a shorter effective life, and a life set too long lowers the annual charge, so n, r and S belong in sensitivity analysis when capital is a large share of cost.

Worked examples

• Retinal camera with a GBP 5,000 resale value annualised over five years

  An illustrative retinal camera costs GBP 50,000, lasts 5 years and has a resale value of GBP 5,000, with costs discounted at 3.5%. The annuity factor is about 4.5151 and the present value of the resale value about GBP 4,210, so the annualised charge is about GBP 10,142 a year, against GBP 9,000 under straight-line depreciation.

  K = 50000; S = 5000; r = 0.035; n = 5; A = 4.51505; E = 10141.66

Common errors

• Subtracting an undiscounted resale value before annualising

  Netting the full GBP 5,000 resale value off the camera's price, rather than its present value of about GBP 4,210, gives about GBP 9,967 a year instead of GBP 10,142. The shortfall is r times S, GBP 175, the yearly cost of the funds still tied up in the resale value.

• Nominal annualisation rate applied to real prices

  Annualising with a nominal rate, which includes expected inflation, while prices are held in one price year overstates the opportunity cost of capital and so the annual charge. The rate and the prices are both real or both nominal.

• Purchase price and annualised charge counted for the same asset

  A model that enters the purchase price in the year of purchase and also adds an annualised charge for the same asset counts the capital twice. The same applies to published unit costs that already include an annualised building element.

Sources

• WHO guide formula for annualising capital net of resale value

  Tan-Torres Edejer T, Baltussen R, Adam T, Hutubessy R, Acharya A, Evans DB, Murray CJL, editors. Making choices in health. WHO guide to cost-effectiveness analysis. Geneva, World Health Organization, 2003. Section 3.3.2 on the annualisation of capital investments, which nets the discounted resale value off the purchase value, divides the result by the annuity factor A(n, r), and uses the replacement cost and a full life for old equipment.

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• Green Book treatment of residual value used for the resale term

  HM Treasury. The Green Book (2026). London, HM Treasury, updated 5 February 2026. Section on assets, which states that an asset's residual value at the end of the appraisal period is included to reflect its opportunity cost and does not depend on an actual sale.

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• GHCC reference case on replacement prices and resold capital

  Global Health Cost Consortium. Reference Case for Estimating the Costs of Global Health Services and Interventions. 2017. Principles 11 and 12, which state that economic evaluations may need replacement prices to capture current opportunity cost and that capital bought some years ago can still be resold and so still has an opportunity cost.

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Canonical Identity