Graphic Chart—Annuity Method
In the above chart the points A¹, A², etc., represent the appraised values plus interest, the segments, A¹I¹, A²I², etc., representing interest on each period’s investment. It is to write down these increases in value that the additional $28 of periodic depreciation charges is needed. The curve OC is a straight line since each period’s depreciation charge is the same. Curves AB and OD are identical with those of the sinking fund method. Curve OE represents the accumulating interest on investment as shown by the segments A¹I¹, A²I², etc. Curve OC is the sum or resultant of curves OD and OE.
The annuity method is termed the “Equal Annual Payment” method in a preliminary report of the Valuation Committee of the American Society of Civil Engineers. As here illustrated it does bring about an equal periodic charge but only because the assumed rate of interest for the sinking fund accumulations is taken also as the rate for interest on the investment. If these two rates differ, the periodic charges will also differ. For example, if the sinking fund rate is taken as 5% and the rate applicable to the appraised values is 8%, the sum of these two interest amounts will not be constant because the bases on which they are calculated are changing each period. Because of this fact the Committee above referred to called this the “Compound Interest” method in its final report. To distinguish this from the sinking fund method which also uses the compound interest principle, the title here adopted, i.e., the “Annuity” method seems to accomplish that purpose.
(c) Unit Cost Method
A third method which uses the compound interest principle is called the “Unit Cost” method. Because of the involved mathematical processes required for the calculation of the amount of its periodic charge, and the doubtful practical value of the method, only a description of its main features will be given here.[33] The aim of this method is to equalize over each unit of product three costs, viz.: the cost of interest on investment, the cost of operation and repairs, and the true depreciation cost, all of these to be included in a periodic charge under the title of “depreciation.”
The calculation of the true depreciation cost by the sinking fund principle is the reason for including this method in the compound interest type. The problem to be solved is the determination of the price to be paid for an asset at a given time so that the cost of each unit of product turned out during its remaining service life shall be the same as the cost of each unit of product turned out during the spent portion of its service life. The difference between the original cost of the asset and the price determined as above will be the depreciation of the asset for the elapsed period. A symbolic showing of the problem will make the matter clear. The following notation will be used:
| V | = | original cost of the asset installed ready for use |
| V₁ | = | price that could be paid for it at the end of the |
| period as above explained | ||
| O | = | estimated average operating costs per period, |
| including repairs, for V | ||
| o | = | estimated average operating costs per period, |
| including repairs, for V₁ | ||
| D[34] | = | true depreciation rate or multiplier under |
| the sinking fund method, for V | ||
| d[35] | = | true depreciation rate or multiplier under |
| the sinking fund method, for V₁ | ||
| U | = | units of output for V during one period |
| u | = | units of output for V₁ during one period |
| r | = | rate of interest on the investment |
Then:
| O + DV + Vr | = the cost per unit of output for V, |
| U | |
| o + dV₁ + V₁r | = the cost per unit of output for V₁. |
| u | |
Since, by hypothesis, these two costs are to be equal, we may form the equation