The two other arbitrary types of this variable percentage method are hardly to be classed as methods as they do not rest on any law according to which they may be operated. Under them arbitrary amounts are charged to depreciation each period, the only controlling principle being that in the one case these periodic amounts increase from period to period, while in the other case they decrease. In the one case, therefore, the appraisal schedule would be similar as to its “Periodic Depreciation” column to those of the two methods just explained, excepting that the column must be reversed, i.e., read from the bottom upward. In the other case, the appraisal schedule would be exactly similar to those just shown. Within the restriction that they must be increasing or decreasing amounts for succeeding periods and that the total depreciation must be charged off within the estimated life-period of the asset, the periodic depreciation charges are, under these methods, purely arbitrary, neither based on fact nor logic.

3. Compound Interest Methods

The third general type of methods for making the depreciation estimate may be called the “Compound Interest” type. This differs radically from any of the others in that it uses the compound interest principle to determine the amount of periodic depreciation. In the practical application of some of these methods, not only is depreciation estimated on this basis but an actual fund of cash or other assets is set aside for accumulation on the compound interest principle so as to provide ready funds for financing the replacement when the old asset is discarded. The setting aside of the fund is not an essential part of the method, and its discussion is therefore deferred to Chapter XXV where the subject of funds and their treatment is fully considered. Under this type there are three methods:

(a) Sinking Fund Method

The problem under the “Sinking Fund” method is the calculation of the amount of a sum of money which placed at compound interest at the end of successive periods will accumulate to the amount of the total depreciation of the asset during its life-term. At the end of each period, this amount plus any accumulated interest on the amounts previously set aside becomes the depreciation estimate for this period. Unless a fund is actually established, there can, of course, be no accumulation of interest. Under this method the amount of such theoretically accumulated interest is, nevertheless, made a part of the periodic depreciation charge.

The method thus becomes simply a mathematical device for making the estimate. The calculation of the fixed periodic amount is in accordance with the following mathematical formula, using the notation given on page 151. The development of this formula is given in full in [Chapter XXV], “The Sinking Fund”:

(3) A = (V - Vₙ)r = Dr
Rⁿ - 1Rⁿ - 1

For the asset used in the other illustrations, i.e., for an asset with a full valuation at the beginning of its life of $150 and a residual value of $50 after a service life of five years, this periodic amount is $18.10, if interest is reckoned at 5%. Any other rate of interest within reason would, of course, be equally appropriate.

The following appraisal schedule may be set up: