Accounting theory and practice, Volume 2 (of 3) : a textbook for colleges and…

The second main classification of methods, called for want of a better title, “Variable Percentage” methods, differs from the proportional methods in that either the base or the percentage rate varies for each estimate of depreciation. The various proportional methods can all be expressed as percentages but their base remains fixed and is always the total amount of depreciation to be charged off. Under the variable percentage methods, if the percentage is fixed, the base varies; and if the base is fixed, the percentage varies. The subclasses here are: (a) Fixed Percentage of Diminishing Value Method (b) “Changing Percentage of Cost Less Scrap” Method (sometimes known as the “Sum of Expected Life-Periods” Method) (c) Arbitrary with Increasing Amounts (d) Arbitrary with Decreasing Amounts (a) Fixed Percentage of Diminishing Value Method The “Fixed Percentage of Diminishing Value” method estimates the periodic depreciation as a fixed percentage of the appraised or book value of the asset as at the time of the last appraisal. Thus, if the asset cost $1,000 and the fixed rate is 10%, the first depreciation estimate is $100 (10% of $1,000) giving an appraised value of $900; the second depreciation estimate is $90 (10% of $900), with a new appraised value of $810; the third estimate is $81 (10% of $810), with an appraisal of $729 for the asset; and so on. It is evident that a final zero valuation can never be reached (although it may be approximated) as the series becomes an indefinite or indeterminate series. If there is any scrap value, and there usually is, the series becomes determinate. From the standpoint of calculation the problem here is the determination of the fixed rate necessary to reduce the asset value to residual or scrap value in the given life-term. Using the standard notation, we may formulate the following equations: V₁ = V(1 - d); V₂ = V₁(1 - d) = V(1 - d)(1 - d); V₃ = V₂(1 - d) = V(1 - d)(1 - d)(1 - d); whence Vₙ = V(1 - d)ⁿ, which solved for 1 - d gives ______ 1 - d = ⁿ√Vₙ/V) , and, solving for d, we get _____ (2) d = 1 - ⁿ√(Vₙ/V) While complex, the formula is readily solvable by means of logarithms. For an asset costing $150 with a service life of 5 years and a scrap value of $50, the rate is found by the above formula to be approximately 19.726%. _______ d = 1 - ⁵√(50/150) = .19726 The appraisal schedule is, therefore, as follows: =======+==============+============+=================+============== | Fixed | | | Total Age in | Depreciation | Periodic | Depreciated or | Accumulated Periods| Rate % |Depreciation| Appraised Value | Depreciation -------+--------------+------------+-----------------+-------------- 0 | ..... | $ ..... | $150.00 | $ ..... 1 | 19.726 | 29.59 | 120.41 | 29.59 2 | 19.726 | 23.75 | 96.66 | 53.34 3 | 19.726 | 19.07 | 77.59 | 72.41 4 | 19.726 | 15.32 | 62.27 | 87.73 5 | 19.726 | 12.27 | 50.00 | 100.00 | | ------ | | | | 100.00 | | -------+--------------+------------+-----------------+-------------- The following chart shows graphically the appraised values and the accumulated depreciation: [Illustration: _Graphic Chart—Fixed Percentage of Diminishing Value Method_] (b) Changing Percentage of Cost Less Scrap Method Similar in effect to the method just explained is the “Changing Percentage of Cost Less Scrap” or the “Sum of Expected Life-Periods” method. Here, the base remains fixed, but the periodic depreciation rates change. Each depreciation rate is a fraction the common denominator of which is the sum of the expected life-periods as viewed from the beginning of each successive period, and the numerator of which is the expected life for the period in question. For example, an asset of which the expected life is 5 periods has at the beginning of each successive period expected life-terms of 4, 3, 2, and 1 periods respectively, making a total of 15 which becomes the common denominator of the fractions whose numerators are 5, 4, 3, 2, and 1 respectively; i.e., the changing depreciation rates are ⁵/₁₅, ⁴/₁₅, ³/₁₅, ²/₁₅, and ¹/₁₅. For an asset costing $150 with expected life of 5 periods and scrap value of $50, the appraisal schedule would be as follows: =========+==============+==============+==============+============= Age | Changing | | Depreciated | Total in | Depreciation | Periodic | or Appraised | Accumulated Periods | Rate % | Depreciation | Value | Depreciation ---------+--------------+--------------+--------------+------------- 0 | ..... | $ ..... | $150.00 | $ ..... 1 | 33⅓ | 33.33 | 116.67 | 33.33 2 | 26⅔ | 26.67 | 90.00 | 60.00 3 | 20 | 20.00 | 70.00 | 80.00 4 | 13⅓ | 13.33 | 56.67 | 93.33 5 | 6⅔ | 6.67 | 50.00 | 100.00 ---------+--------------+--------------+--------------+------------- A comparison of this appraisal schedule with that of the fixed percentage of diminishing value method shows that this method charges more depreciation during the early life-periods and less during the later periods. The general effect of this method and its significance are discussed in Chapter X where the relative merits of the various methods are considered. The graph for the sum of expected life-periods method is not shown as it differs little from that of the fixed percentage of diminishing value method on page 158. (c, d) Arbitrary Methods 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.