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 (b) Annuity (c) Unit Cost (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”: (V - Vₙ)r Dr (3) A = --------- = ------ Rⁿ - 1 Rⁿ - 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: =======+==================================+============+============ |Periodic Depreciation Composed of:| | Age in +-----------+----------+-----------+Depreciated | Total Periods| Periodic | Interest | Total |or Appraised|Accumulated | Amount | | | Value |Depreciation -------+-----------+----------+-----------+------------+------------ 0 | $ ..... | $ .... | $ ...... | $150.00 | $ ...... 1 | 18.10 | .... | 18.10 | 131.90 | 18.10 2 | 18.10 | .90 | 19.00 | 112.90 | 37.10 3 | 18.10 | 1.85 | 19.95 | 92.95 | 57.05 4 | 18.10 | 2.85 | 20.95 | 72.00 | 78.00 5 | 18.10 | 3.90 | 22.00 | 50.00 | 100.00 | | | ------- | | | | | $100.00 | | -------+-----------+----------+-----------+------------+------------ The following chart shows graphically the appraised values, the accumulating depreciation and the elements which compose it, the curve OD representing the fixed periodic amounts, and EF the theoretical interest accumulations. The curve OD is a straight line inasmuch as it represents fixed periodic amounts. The depreciation and interest curves, OC and EF, representing gradually increasing amounts, are both slightly concave and would become increasingly so the longer the period covered. The appraisal curve AB is slightly convex and its convexity is accelerated by lapse of time. [Illustration: _Graphic Chart—Sinking Fund Method_] (b) Annuity Method The “Annuity” method also makes use of the compound interest principle, but in addition to the method of the sinking fund it adds to the periodic depreciation charge as determined thereunder interest on the successive appraised values of the asset. The effect of this is to charge to the product, by way of Profit and Loss, interest on the capital invested in each asset used in manufacture. The appraised values of the asset are exactly the same as under the sinking fund method, but the expense charge to depreciation is larger under the annuity method by the interest on the appraised value of the asset. This charging of interest to the product under the title “depreciation” makes it necessary to capitalize the interest charge by adding it to the value of the asset. If the amount to be charged off, i.e., V-Vₙ is the same under both methods, for both to arrive at the same scrap value, Vₙ, the interest under the annuity method must be added to the value of the asset each time before deducting the depreciation charge, a part of which is this same interest. The annuity method thus makes a larger periodic charge than the sinking fund method. The problem involved in the calculation of the periodic depreciation charge by the annuity method is sometimes stated as the method of finding a fixed or constantly equal periodic charge sufficient to charge off not only depreciation as such but also the interest which has been added to the value of the asset. The mathematical formula may be derived as follows, using the standard notation: VR = V(1 + r), or the asset with interest added to it VR - D₁ = V₁, appraised value at end of first period V₁R - D₁[31] = VR² - D₁R - D₁ = V₂, appraised value at end of second period V₂R - D₁[32] = VR³ - D₁R² - D₁R - D₁ = VR³ - D₁(R² + R + 1) = V₃, appraised value at end of third period, etc. [31] Inasmuch as by hypothesis the periodic depreciation charges are the same D₁ = D₂ = D₃ etc., hence, D₁ is used to represent the fixed depreciation charge per period. [32] Inasmuch as by hypothesis the periodic depreciation charges are the same D₁ = D₂ = D₃ etc., hence, D₁ is used to represent the fixed depreciation charge per period. Generalizing, we have: Vₙ₋₁R - D₁ = VRⁿ - D₁(Rⁿ⁻¹ + Rⁿ⁻² ... + R² + R + 1) = Vₙ, scrap value. Whence Rⁿ - 1 VRⁿ - Vₙ = D₁ -------. Solving for D₁; we have: R - 1 (VRⁿ - Vₙ)(R - 1) (VRⁿ - Vₙ)r (4) D₁ = ----------------- = ------------, Rⁿ - 1 Rⁿ - 1 = periodic depreciation charge That this is the same as the amount of the periodic depreciation charge by the sinking fund method plus interest on the investment, is seen by comparing formula (4) with formula (3). Formula (4) may be written as (VRⁿ - Vₙ)r + Vr - Vr ---------------------- , Rⁿ - 1 i.e., the quantity Vr-Vr = 0 is put into the numerator. Adding zero (Vr-Vr = 0) cannot change its value. Performing the multiplication indicated by the parentheses, we have VRⁿr - Vₙr + Vr - Vr --------------------- . Rⁿ - 1 Rearranging the terms, we have (Vr - Vₙr) + (VRⁿr - Vr) ------------------------, Rⁿ - 1 which factored in each group becomes (V - Vₙ)r Vr(Rⁿ - 1) -------- + ---------- . Rⁿ - 1 Rⁿ - 1 By reducing the second fraction, this may be written as: (V - Vₙ)r (5) --------- + Vr Rⁿ - 1 which is seen to be identical with formula (3) for the sinking fund except for the addition of Vr, which represents interest at r% on the investment V. The identity of the annuity formula (4) with the sinking fund formula (3) plus interest on investment can be established similarly for any of the periods. It will be noted that by the annuity method the whole original value of the investment is always earning interest either in the sinking fund or in the diminishing appraised value. Thus, the portion of original value deducted each period earns interest in the sinking fund, while what is left as appraised value earns interest outside the fund. Thus, the annuity method of charging depreciation may be said to consist of two parts, viz., the fixed periodic amount and interest on the original investment. This will be seen from the appraisal schedule which follows. The same illustrative data are used as before, including interest at 5%. =========+==========================================+ | Periodic Depreciation Charge | | Composed of: | +--------+----------+------------+---------+ | | | | | Age in | | Interest | Interest | | Periods | Fixed | on | on | Total | | Amount | Fixed | Investment | Charge | | | Amount | | | | | | | | | (a) | (b) | (c) | (d) | ---------+--------+----------+------------+---------+ | | | | | 0 | $..... | $.... | $..... | $...... | 1 | 18.10 | | 7.50 | 25.60 | 2 | 18.10 | .90 | 6.60 | 25.60 | 3 | 18.10 | 1.85 | 5.65 | 25.60 | 4 | 18.10 | 2.85 | 4.65 | 25.60 | 5 | 18.10 | 3.90 | 3.60 | 25.60 | +--------+----------+------------+---------+ | $90.50 | $9.50 | $28.00 | $128.00 | ---------+--------+----------+------------+---------+ =========+===========+==============+========================= | | | Accumulated | | | Depreciation | | +-----------+------------- | Appraised | Depreciated | | Age in | Value | or Appraised | | Periods | Plus | Value | Including | True | Interest | | Interest | Depreciation | | | | | | | | | (e) | (f) | (g) | (h) ---------+-----------+--------------+-----------+------------- | | | | 0 | $...... | $150.00 | $...... | $...... 1 | 157.50 | 131.90 | 25.60 | 18.10 2 | 138.50 | 112.90 | 51.20 | 37.10 3 | 118.55 | 92.95 | 76.80 | 57.05 4 | 97.60 | 72.00 | 102.40 | 78.00 5 | 75.60 | 50.00 | 128.00 | 100.00 | | | | | | | | ---------+-----------+--------------+-----------+------------- It will be noted, as stated above, that the sum of the two interest items in columns (b) and (c) is the same for each period, viz., $7.50, interest on the _original_ investment. The total of column (d), total charge for periodic depreciation, minus the total of column (c), interest on the diminishing appraised values, gives $100, the true depreciation as shown by column (h). _True_ depreciation under the annuity method is the same as under the sinking fund method. The periodic depreciation charge differs, however. A graphical illustration of the main items of the appraisal schedule is shown below: [Illustration: _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.” [33] The interested student is referred to Proceedings—American Society of Civil Engineers—December, 1916, in which will be found the Valuation Committee’s report, and to Salier’s “Principles of Depreciation,” in both of which the formula is developed. 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 [34] D and d may be expressed as the respective periodic amount-multipliers necessary to create sinking funds of one dollar each under the conditions as to time and rate for V and V₁. VD and V₁d become, therefore, the periodic amounts of true depreciation, i.e., decrement in value. [35] D and d may be expressed as the respective periodic amount-multipliers necessary to create sinking funds of one dollar each under the conditions as to time and rate for V and V₁. VD and V₁d become, therefore, the periodic amounts of true depreciation, i.e., decrement in value. 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 O + DV + Vr o + dV₁ + V₁r ------------- = -------------, U u which solved for V₁, the price to be paid, gives u ( ) --- ( O + DV + Vr ) - o U ( ) V₁ = ------------------------ . (d + r) Evidently, V-V₁ is the amount of the depreciation charge. The values for D and d, as indicated in the footnote, may be substituted and the amount of V₁ determined. The process is somewhat complicated in its practical application and will not be carried further here.