Functions & Formulae

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

Capital outlay annualisation function at the evaluation discount rate

a(K, S, r, n) = E

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.

  • Annualised capital charge net of discounted resale value

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

    Divides the purchase price, net of the resale value discounted over the useful life, by the annuity factor A, the present value of one unit of currency received at the end of each of n years. The result is the economic cost of using the asset for one year, covering both the value used up and the opportunity cost of the funds tied up in it. Rearranged, E equals K minus S divided by A, plus r times S, so the resale value attracts only a yearly holding charge of r times S and no wear.

  • Annualised capital charge with no resale value

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

    The special case of HE-FM-ANLS-001 with S equal to zero, used for inputs that have no value at the end of their life, such as training that has to be repeated. The purchase price is divided by the annuity factor, so the charge works like a loan repayment at rate r, paying a return on the balance still tied up and recovering part of the price each year.

  • Ratio of the annualised capital charge to straight-line depreciation

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

    With no resale value, the annualised charge, K divided by A, and the straight-line charge, K divided by n, differ by the ratio n divided by A, which depends only on the useful life and the rate. Because A is smaller than n at any positive rate, the ratio exceeds 1 and rises with the useful life, and it tends to 1 as r approaches zero.