Signature
A = (1 - (1+r)^(-n)) / r; R_SL = n / A
| Inputs | Definition | Unit |
|---|---|---|
r | Annual discount rate, entered as a proportion greater than zero | proportion per year |
n | Years of service of the asset | years |
A | Present value of one unit of currency received at the end of each of n years at rate r | dimensionless |
|---|---|---|
R_SL | Equivalent annual cost with no resale value divided by the straight-line charge, the price divided by the useful life | dimensionless |
Function
Capital outlay annualisation function at the evaluation discount rate
Maps a one-off capital outlay to the equal annual sum charged in each year of the asset's useful life. Its inputs are the purchase price K, the resale or residual value S, the annual discount rate r and the useful life n, and its output is the equivalent annual cost E, set so that the present value of E over n years equals K less the present value of S. It is the inverse of present-value discounting at the same rate (HE-FN-DR-001), applied to a single capital input in economic costing, and it moves no money, unlike the annuity payment contract of HE-FN-AP-001. The same equation appears in the Cost package as HE-FM-COST-003; this package adds the zero-resale form, the comparison with straight-line depreciation and the year-by-year split of the charge. Notation follows the Annualisation article.
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Implementations
Excel
Annualised to straight-line ratio from named cells
With the life in UsefulLife and the rate in DiscountRate, the formula returns R_SL. Multiplying it by the straight-line charge gives the equivalent annual cost with no resale value.
=UsefulLife/((1-(1+DiscountRate)^-UsefulLife)/DiscountRate)
Assumptions
Straight-line and annualised charges with no resale value
Both charges use the same price and useful life and no resale value. With a resale value the straight-line charge is K minus S divided by n, and the gap also includes the holding charge of r times S.
Worked examples
Five-year device ratio of annualised to straight-line charge at 3.5%
For a five-year device at 3.5% the annuity factor is about 4.515, so the economic charge is about 11% above straight-line depreciation, as in the article's table.
n = 5; r = 0.035; A = 4.51505; R_SL = 1.1074
Sixty-year building ratio of annualised to straight-line charge at 3.5%
For a building with a 60-year life at 3.5% the annuity factor is about 24.945, so the economic charge is about 2.4 times the straight-line charge. Services that occupy purpose-built space show the largest effect.
n = 60; r = 0.035; A = 24.94473; R_SL = 2.4053
Sixty-year building ratio of annualised to straight-line charge at 1.5%
At 1.5%, the rate NICE allows for alternative analyses in specific circumstances (PMG36, section 4.5.2), the annuity factor for 60 years is about 39.380 and the economic charge about 1.52 times the straight-line charge, so the gap for long-lived assets depends heavily on the rate.
n = 60; r = 0.015; A = 39.38027; R_SL = 1.5236
Common errors
Treating the annualisation gap as small for buildings
The 11% gap for a five-year device at 3.5% does not carry over to long-lived assets. For a 60-year building the economic charge is about 2.4 times the straight-line charge, so straight-line depreciation understates building costs far more than equipment costs.
Sources
GHCC reference case on straight-line depreciation against an annualisation factor
Global Health Cost Consortium. Reference Case for Estimating the Costs of Global Health Services and Interventions. 2017. Principle 12, which states that financial cost estimates use straight-line depreciation and economic costs use an annualisation factor that adjusts the years of life for opportunity cost by means of a discount rate.
Turner and colleagues on the annualisation factor being smaller than the lifespan
Turner HC, Sandmann FG, Downey LE, Orangi S, Teerawattananon Y, Vassall A, Jit M. What are economic costs and when should they be used in health economic studies? Cost Effectiveness and Resource Allocation. 2023, volume 21, article 31. Figure 1, which notes that the annualisation factor is smaller than the expected lifespan, 4.58 against 5 in its example, so the annualised economic cost exceeds the annualised financial cost, even from a provider or payer perspective.
Canonical Identity
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