Accounting theory and practice, Volume 2 (of 3)

Methods of Calculation

Many methods of reckoning depreciation have been devised; some good, some bad, and some too theoretical and involved ever to serve the practical needs of business. It is proposed here to explain those methods which have found most favor and, in Chapter X, to discuss their effects from the standpoint of the true purpose of any scheme of depreciation.

There are several ways in which the depreciation calculation can be made, and the methods may be classified broadly, though with some overlapping, under four heads. These classes are:

• 1. Proportional Methods
• 2. Variable Percentage Methods
• 3. Compound Interest Methods
• 4. Miscellaneous Methods

Factors of Calculation

The factors which must be known under most methods of calculation are:

• 1. Original cost of the asset
• 2. Scrap or residual value
• 3. Estimated service life
• 4. Under some methods, an arbitrary interest rate

By original cost is meant full cost of the asset in position ready for use. By scrap value is meant the estimated value of the asset at the time of its discard, when removed from position and ready for sale or other mode of disposal. This is sometimes called its salvage value as distinguished from its value while still in position to render service but awaiting discard or break-up. By estimated service life is meant the time during which the asset will be used for service. This may be expressed in ordinary units of calendar time, such as the year; or in units of service time, such as working hours; or finally in units of output, such as tons, cubic feet, kilowatt hours, etc.

Symbols to be Used

For purposes of notation and reference the following symbols will be used:

• V = original cost
• V₁ = depreciated value at end of first period
• V₂ =      ”        ”  ”  ”  ”  second period
• Vₙ =      ”        ”  ”  ”  ”  last period, i.e., scrap value
• n  = length of service life
• d  = rate of depreciation per period
• D  = total amount of depreciation, i.e., V - Vₙ
• D₁ = amount of depreciation during first period
• Dₙ =  ”    ”      ”        ”    last    ”
•  r = rate of interest
• R = 1 + r, i.e., 100% plus the rate
• 1 - r = 100% minus the rate
• A = annual amount to be paid under the compound interest methods

It should be noted that the foregoing factors are all estimates, with the exception of original cost, and that when making these estimates the principles of depreciation as discussed in preceding chapters must be taken into consideration. Thus, a decision must be made as to whether physical or functional depreciation is the controlling factor in determining service life. In the determination of service life, the policy as to repairs, renewals, and maintenance has a very important bearing. In fixing the scrap value, the relation of inadequacy, obsolescence, wear and tear, and age to the values remaining in the asset must be considered.

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:

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:

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.

(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.

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.

(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.

2. Variable Percentage Methods

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

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:

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%.

The appraisal schedule is, therefore, as follows:

The following chart shows graphically the appraised values and the accumulated depreciation:

(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:

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.

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”:

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:

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.

(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.

Generalizing, we have:

• Vₙ₋₁R - D₁ = VRⁿ - D₁(Rⁿ⁻¹ + Rⁿ⁻² ... + R² + R + 1) = Vₙ,
• scrap value. Whence

Solving for D₁; we have:

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

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

Rearranging the terms, we have

which factored in each group becomes

By reducing the second fraction, this may be written as:

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%.

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:

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:

Then:

Since, by hypothesis, these two costs are to be equal, we may form the equation

which solved for V₁, the price to be paid, gives

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.

4. Miscellaneous Methods

Other methods of calculating the depreciation charge are used, but they cannot be classified under any of the three groups discussed so far. They are a miscellaneous, mongrel breed, scarcely to be dignified in some instances as methods. Among these may be mentioned the following:

• (a) Maintenance Method
• (b) Replacement Method
• (c) Fifty Per Cent Method
• (d) Appraisal Method
• (e) Insurance Method
• (f) Gross Earnings Method

(a) Maintenance Method

In this case a periodic charge for depreciation is made, equal in amount to the cost of maintenance of the asset for the period. It is thus a definite but variable amount, depending upon the maintenance policy.

(b) Replacement Method

This is hardly a method of calculating the depreciation charge, but rather of recognizing the fact of depreciation by charging all renewals and replacements to revenue. It is argued that in a large, widely extended plant after depreciation has reached the point where renewals are necessary, the charging of all renewals and replacements as expenses will take care of all accruing depreciation and secure a fairly uniform charge to product from period to period. Under this plan depreciation as such does not appear on the books but is taken care of under other titles.

(c) The Fifty Per Cent Method

This is somewhat similar to the replacement method in that it is applicable only after depreciation has reached the renewals stage. It is claimed for it that, in a property or class of asset consisting of many similar parts, as railroad ties, for example, after the stage of normal repairs has been reached so that the parts are in all degrees of repair from 0% to 100%, the normal maintenance and renewals policy will maintain the property or asset always in about 50% condition. Therefore the total depreciation for the asset or class is the other 50%, which never reaches a larger amount because of a constant renewal of parts. This 50% depreciation may or may not be carried on the books but it exists nevertheless. For the conditions under which it is applicable as above, the law of averages doubtless applies and makes the estimate a fairly good one.

(d) Appraisal Method

Here a physical appraisal of the asset or property is taken at the close of every fiscal period. The difference in value between the two appraisals for successive fiscal periods represents the depreciation for the period and would be brought on the books as such.

(e) Insurance Method

This is applicable only to large properties with assets widely distributed. Its operation “involves the actuarial principles of ordinary insurance. This means that the fund accumulated by depreciation charges should not be reserved as an accumulation until it can be spent for the purpose of replacing the identical property upon which the fund accumulated when such property is abandoned; and furthermore, that this fund should be expended, in whole or in part, during the year in which it is created, in the replacement of equipment.”

(f) Gross Earnings Method

Here the depreciation estimate is based on the gross earnings for the period. This does not necessarily mean that the depreciation estimate will be large when profits are large, and small or nothing when profits are small, although it may be made to apply in that way in individual cases. The policy of making ample reserves for depreciation in good years and scant reserves in poor years is not to be wholly condemned. Depreciation, however, has no relation to, or dependence upon, profits. Rather, profits depend on depreciation in the sense that they cannot exist until after charges for depreciation have been taken care of. Depreciation considered as a fixed per cent of gross earnings is almost the same in effect as the service output method, and has much to commend it.

Condition Per Cent

Before leaving the topic of method, it may be well to explain a term used in connection with the depreciation estimate, viz., condition per cent. The condition per cent of an asset is found by subtracting from 100%, the fraction which represents the ratio of the present accumulated depreciation to the total estimated depreciation. Thus, if an asset has depreciated in value one-quarter, its condition per cent is said to be 75 (100%-25%). Hence, condition per cent is easily calculated if depreciation has been estimated by any of the proportional methods. If, in addition to the standard notation used, we assume that:

• Dₘ = total amount of depreciation for m periods
• Vₘ = value of the asset at end of m’th period

then, in general, condition per cent may be expressed by the formula:

Evidently, therefore,

Under the proportional methods Dₘ/D = nd. Therefore, condition per cent is 100%-nd.

Under the sinking fund method, the calculation is more complex. Dₘ, the total amount of depreciation accumulated to date, i.e., after m periods, is the amount of the annuity A for m periods. From formula (3), Chapter XV, page 272, the amount of an annuity A is seen to be

Therefore,

from which the ratio

Accordingly, condition per cent under the sinking fund method is: