1. PROPORTIONAL METHODS

Proportional methods include all those in which the periodic depreciation is calculated as a proportional part of some fixed basic value. These may be grouped under the following subclass titles, each of which will be separately considered: (a) Straight Line (b) Working Hours (c) Composite Life (d) Service Output (a) Straight Line Method The “Straight Line” method, so called because its graphical representation is a straight line, is perhaps the simplest of all methods for calculating the periodic depreciation charge and is therefore the most widely used. Under it the loss of value for each period is proportioned to the length of service life. Thus, an asset whose service life is reckoned as 10 years will have depreciated one-tenth by the end of the first year, two-tenths by the end of the second year, and so on. The depreciation for each year is the same. Shown by formula, using the notation above, this will be: (V - Vₙ) D (1) D₁ = ------- or --- and, evidently, n n V₁ = V - D₁; V₂ = V₁ - D₂; etc. From these formulas a schedule of appraisal for a given asset may be made up showing its values as at the end of each year of its estimated life. For an asset costing $150, of which the service life is 5 periods and the scrap value, due to inadequacy, is $50, such an appraisal schedule would work out as follows: ==========+==============+==================+================== Age in | Periodic | Depreciated or | Total Accumulated Periods | Depreciation | Appraised Value | Depreciation ----------+--------------+------------------+------------------ 0 | $..... | $150.00 | $..... 1 | 20.00 | 130.00 | 20.00 2 | 20.00 | 110.00 | 40.00 3 | 20.00 | 90.00 | 60.00 4 | 20.00 | 70.00 | 80.00 5 | 20.00 | 50.00 | 100.00 ----------+--------------+------------------+------------------ Here, the fixed depreciation base, i.e., the amount to be written off over the 5 periods, is $100 ($150-$50). Graphically presented below, we have the straight line AB representing the periodic appraised values, and the line OC, the accumulating depreciation. [Illustration: _Graphic Chart—Straight Line Method_] (b) Working Hours Method Where the “Working Hours” method is used, the life of the asset, instead of being estimated in calendar units of time, is given in service units as so many working hours. Thus, if it is estimated that a machine will stand 12,000 hours of operation, its service life is stated as 12,000 working hours. Record is kept of the number of hours the machine is operated during each fiscal period and compared with the estimated length of service life (also in working hours) to give the proportion of the total depreciation which must be charged to a given period. Using the same example—an asset costing $150, with scrap value of $50, and service life of 12,000 working hours—the appraisal schedule will be as shown below, assuming that during the first fiscal period the asset was used 3,000 hours, during the second 4,500 hours, the third 2,700, the fourth 1,200, and that it was scrapped some time during the fifth fiscal period after another 600 hours of operation. =========+=========+==============+=================+============= | | | | Total Age in | Service | Depreciation | Depreciated or | Accumulated Periods | Hours | Rate % | Appraised Value | Depreciation ---------+---------+--------------+-----------------+------------- 0 | ..... | ..... | $150.00 | $...... 1 | 3,000 | 25 | 125.00 | 25.00 2 | 4,500 | 37½ | 87.50 | 62.50 3 | 2,700 | 22½ | 65.00 | 85.00 4 | 1,200 | 10 | 55.00 | 95.00 5 | 600 | 5 | 50.00 | 100.00 | ------ | ------- | | | 12,000 | 100 | | ---------+---------+--------------+-----------------+------------- The following chart shows graphically the appraised values and the accumulated depreciation as at the end of successive fiscal periods. The character of the curves here has no relation to the lapse of calendar time, i.e., fiscal periods, but depends entirely on the degree of intensity of operation of the asset, i.e., its service life in working hours. The above curves are true, therefore, only for the particular data assumed and do not, in any way, indicate a characteristic tendency of this method. Were the bottom line of the chart, the abscissa, laid off proportionately on the basis of working hours instead of fiscal periods, both graphs would, of course, become straight lines. It is only because information as to values is desired at the close of each fiscal period that the curves representing values become broken lines. [Illustration: _Graphic Chart—Working Hours Method_] (c) Composite Life Method Another proportional method, which in its operation is similar to the straight line method, is known as the “Composite Life” method. Its feature is the calculation of depreciation on the plant as a whole, rather than on each individual asset. Under it what is known as the _mean_ life of the plant is calculated. Depreciation may then be estimated, on the straight line or other basis, for the entire plant. To determine mean life it is necessary to “weight” the life of each individual asset with its value and so get a common basis, the dollar-year, for all assets. The process of calculating mean life will be explained in Chapter XI where also its use and adaptability are discussed. Aside from the determination of mean life, the method does not differ from others which have been or will be discussed. (d) Service Output Method A fourth type of proportional method is known as the “Service Output” method. Under it, the life of the asset is reckoned in terms of _quantity_ of output. This is very similar to the working hours method but differs in the unit of measurement for service life. The life of the asset is measured by its product. Thus, the life of a water filter may be given in terms of gallons or cubic feet of water run through it; that of a rock crusher in terms of the cubic feet of rock handled; that of a freight car or engine in terms of car miles; and so on. If, therefore, record is kept of the performance of such an asset for the fiscal period, the amount of depreciation to be charged off is readily calculated, being such a portion of the total depreciation as the units of output for the current period bear to the total units of life output. It is thus a straight proportional method. Its appraisal schedule and graphical representation are exactly similar to those of the working hours method. It should be said, however, that sometimes the service output method is operated on a sinking fund basis instead of on a total depreciation basis. When the sinking fund basis is used, the total payments into the fund, excluding all interest accretions, constitute the total amount of depreciation to be distributed over the service output.