Annualised capital charge with no resale value

Signature

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

Function

Capital outlay annualisation function at the evaluation discount rate

Computational function

• Computational function for the yearly return on capital and price recovered in an annualised charge

  Takes the price, rate and useful life of an input with no resale value and a year number t, and returns the balance of the price still tied up at the start of that year, the return on it at rate r and the part of the price recovered by that year's charge. It applies HE-FM-ANLS-002 to find E and then splits E as a loan repayment, which shows where the gap between the annualised and straight-line charges comes from.

  Inputs and outputs: K is the purchase price, required, above zero, in currency; r is the annual discount rate, required, above zero, as a proportion; n is the useful life in whole years; t is the year of use, a whole number from 1 to n; A is the annuity factor; E is the annualised charge in currency per year; B_t is the balance still tied up at the start of year t; R_t is the return on that balance, r times B_t; P_t is the part of the price recovered in year t, E less R_t. The last three are in currency.

  Assumption: The input has no resale value, the charge falls at the end of each year and the return is calculated at the rate used to annualise the charge. Over the n years the returns sum to n times E less K and the recovered parts sum to K.

  Worked example (Camera in its first year): In year 1 the full GBP 50,000 is tied up, so the return at 3.5% is GBP 1,750 and the remaining GBP 9,324 of the GBP 11,074 charge recovers part of the price. K = 50000; r = 0.035; n = 5; t = 1; A = 4.51505; E = 11074.07; B_t = 50000; R_t = 1750; P_t = 9324.07

  Worked example (Camera in its final year): In year 5 about GBP 10,700 is still tied up, the return is about GBP 374 and the recovered part, about GBP 10,700, clears the balance, as in the article's table. K = 50000; r = 0.035; n = 5; t = 5; A = 4.51505; E = 11074.07; B_t = 10699.58; R_t = 374.49; P_t = 10699.58

  Excel: =-IPMT(DiscountRate,Year,UsefulLife,CapitalPrice) returns the return R_t for the year in the cell named Year, and =-PPMT(DiscountRate,Year,UsefulLife,CapitalPrice) returns the recovered part P_t. The two sum to the charge from PMT in every year.

  R: capital_schedule <- function(K, r, n) { E <- K*r/(1-(1+r)^(-n)); t <- 1:n; B <- K*(1+r)^(t-1)-E*((1+r)^(t-1)-1)/r; data.frame(year = t, balance = B, return_on_capital = r*B, recovered = E-r*B) } Returns one row per year of use.

  Python: def capital_schedule(K, r, n): E = K*r/(1-(1+r)**(-n)); B = [K*(1+r)**(t-1)-E*((1+r)**(t-1)-1)/r for t in range(1, n+1)]; return [(t, b, r*b, E-r*b) for t, b in enumerate(B, 1)] Returns a list of tuples of year, balance, return and recovered part.

  Test (Recovered parts sum to the price): Over the useful life the recovered parts add up to the purchase price. Expected result TRUE. Excel check =ABS(-CUMPRINC(DiscountRate,UsefulLife,CapitalPrice,1,UsefulLife,0)-CapitalPrice)<0.01

  Test (Returns sum to the total charges less the price): Over 5 years the camera's returns sum to about GBP 5,370, five charges of GBP 11,074 less the GBP 50,000 price. Expected result TRUE. Excel check =ABS(-CUMIPMT(DiscountRate,UsefulLife,CapitalPrice,1,UsefulLife,0)-(UsefulLife*PMT(DiscountRate,UsefulLife,-CapitalPrice)-CapitalPrice))<0.01

  Common error (Charging a return on the full price in every year): Adding r times K, GBP 1,750, to the straight-line charge of GBP 10,000 gives GBP 11,750 a year for the camera, above the correct GBP 11,074, because the balance tied up falls as the price is recovered.

  Source: 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 describes the annualised economic cost as depreciation plus the opportunity cost of the funds tied up in the capital item. The year-by-year split is the standard repayment schedule of an annuity at rate r and follows algebraically from HE-FM-ANLS-002.

  A = (1 - (1+r)^(-n)) / r; E = K / A; B_t = K * (1+r)^(t-1) - E * ((1+r)^(t-1) - 1) / r; R_t = r * B_t; P_t = E - R_t

Try this function

Implementations

• Excel

  Annualised charge with no resale value using Excel PMT

  PMT returns the equal end-of-year payment that repays a loan of K at rate r over n years. The price is entered as a negative present value so that the charge is positive, and the type argument is left at 0 for end-of-year sums.

  =PMT(DiscountRate,UsefulLife,-CapitalPrice)

Assumptions

• No value left at the end of the useful life

  The input is worth nothing when it is replaced. When it can be sold or reused, HE-FM-ANLS-001 with the discounted resale value applies instead.

• Life set by the replacement or repeat interval

  For training and other start-up inputs, n is the interval at which the input is repeated. The GHCC glossary gives the example that training repeated every five years is spread over five years.

Worked examples

• Retinal camera with no resale value annualised over five years

  The GBP 50,000 camera with no resale value, a 5-year life and a 3.5% rate has an annualised charge of about GBP 11,074 a year, against GBP 10,000 under straight-line depreciation.

  K = 50000; r = 0.035; n = 5; A = 4.51505; E = 11074.07

• Screener training annualised over its four-year repeat interval

  Training that costs GBP 12,000 and is repeated every 4 years has an annuity factor of about 3.6731 at 3.5% and an annualised charge of about GBP 3,267 a year, against GBP 3,000 on a straight-line basis.

  K = 12000; r = 0.035; n = 4; A = 3.67308; E = 3267.01

Common errors

• Straight-line depreciation used as an economic annualised charge

  Dividing the GBP 50,000 camera price by 5 years gives GBP 10,000 a year, the financial charge found in provider accounts. It omits the return on the capital tied up in the camera and understates the economic charge of about GBP 11,074 at 3.5% by about 10%.

• Start-of-year PMT type mixed with end-of-year annualisation

  Setting the type argument of PMT to 1 treats each sum as paid at the start of the year and gives about GBP 10,700 a year for the camera instead of GBP 11,074. Mixing that charge with costs discounted at the end of each year understates the capital cost.

Sources

• Turner and colleagues on dividing by an economic annualisation factor

  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 and text, which divide the replacement cost by an annualisation factor based on the expected lifespan and a discount rate for economic costs, against dividing by the lifespan for financial costs.

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• GHCC glossary on spreading repeated training over its interval

  Global Health Cost Consortium. Reference Case for Estimating the Costs of Global Health Services and Interventions. 2017. Glossary entry on start-up costs, which spreads staff training that is repeated every five years over five years.

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