Signature
E = (K - S / (1 + r)^n) / ((1 - (1 + r)^(-n)) / r)
| Inputs | Definition | Unit |
|---|---|---|
K | Cost of acquiring the asset at the start of its life, or its current replacement cost for an existing asset | currency |
S | Value of the asset when it is replaced, zero if it has none | currency |
r | Rate reflecting the opportunity cost of the funds invested, greater than zero | proportion per year |
n | Number of years over which the asset is used before replacement | years |
E | Constant annual cost whose present value over the useful life equals the net capital cost | currency per year in the stated price year |
|---|
Function
Resource costing function
Maps the resources used by an option, and the unit cost of each, to a single cost expressed in one currency, one price year and one time basis. Costing identifies the resources that differ between options, measures how much of each is used and values each with a unit cost; costs from other years are moved to the common price year, and capital used over several years is spread as an equivalent annual cost. Future costs are then discounted, as set out on the Discount Rate page.
Try this function
Implementations
Excel
Equivalent annual cost with the PMT function
With the purchase cost in CapitalCost, the resale value in ResaleValue, the rate in DiscountRate and the life in UsefulLife, Excel returns the equal end-of-year payment whose present value equals the purchase cost less the discounted resale value.
=PMT(DiscountRate,UsefulLife,-CapitalCost,ResaleValue)
Assumptions
Constant annual payments at the end of each year
The annuity factor assumes equal payments at the end of each year of the useful life and a single discount rate. The Unit Costs of Health and Social Care manual depreciates building capital over 60 years at 3.5% declining to 3% after 30 years, which a single-rate factor does not reproduce.
Replacement cost for existing assets
For equipment or buildings already in use, K is the current replacement cost, or the original cost indexed to current prices, with a full useful life, so that assets written off in the accounts still carry an economic cost.
Share attributed to the evaluated service
E is the annual cost of the whole asset. When the asset is shared, the share used by the evaluated service is attributed by use, for example by the proportion of time or throughput, before it enters a cost per patient.
Worked examples
Imaging equipment with no resale value
Equipment costing 500,000 pounds, used for 8 years with no resale value and a 3.5% discount rate, has an annuity factor of about 6.8740 and an equivalent annual cost of about 72,738.32 pounds. The figures are illustrative.
K = 500000; S = 0; r = 0.035; n = 8; E = 72738.32
Same equipment with a resale value
If the equipment can be sold for 50,000 pounds after 8 years, the discounted resale value of about 37,971 pounds is netted off and the equivalent annual cost falls to about 67,214.49 pounds.
K = 500000; S = 50000; r = 0.035; n = 8; E = 67214.49
Common errors
Straight-line depreciation as the annual cost
Dividing 500,000 pounds by 8 years gives 62,500 pounds a year, which omits the opportunity cost of the capital tied up in the asset and understates the equivalent annual cost of about 72,738 pounds at 3.5%.
Sources
Annualisation formula for capital costs
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 Annualization of capital investments (P = K minus S / (1 + r)^n; E = P / A(n,r) with A(n,r) = (1 minus (1 + r)^(-n)) / r; replacement cost for old equipment), attributed to Drummond and colleagues.
Costing textbook on annuitising capital
Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford: Oxford University Press; 2015. Costing chapter: equivalent annual cost of capital using the annuity factor, allowing for resale value.
Annuitised capital in UK unit costs
Jones K, Weatherly H, Barker A, Birch S, Castelli A, Chalkley M, Dargan A, Findlay D, Hinde S, Markham S, Smith D, Teo H. Unit Costs of Health and Social Care 2025 Manual. Canterbury and York: Care and Outcomes Research Centre, University of Kent, and Centre for Health Economics, University of York; 2026. Glossary (annuitising: converting a capital investment into the annual equivalent cost for the period over which it is expected to last) and notes to the service tables (capital costs depreciated over 60 years at a discount rate of 3.5%, declining to 3% after 30 years).
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0