4. Annuity method further considered.—In the preceding chapter, the annuity method was discussed to some extent. It was found that appreciation was calculated at some given rate of interest, and that the shrinkage in the value of the asset amounted to the supposed value of the services realized. In practice, the shrinkages that take place in successive months or years are considered as equal, and constitute an an nuity; hence, the name of this method.
These shrinkages constitute the depreciation, while the appreciations are represented as a source of profit. The annuity rent, or periodic depreciation, must be so chosen that the whole process will reduce the book value of the instrument to its estimated scrap value by the end of its estimated life.
5. Results obtained by annuity method.—The rent of the required annuity, or periodic depreciation, is found, according to the annuity method, by subtract ing from the original cost the discounted value of the estimated scrap value, and dividing the remainder by the present worth of an annuity of one dollar for the required time. But a comparison of this method with that of finding the implied periodic rental value— previously described—reveals the fact that the two are identical.
In other words, according to the annuity method, the annuity, or periodic depreciation, is the periodic rental value itself—that is, it is value of the services when they are received. The net-level annual amount to be written off as depreciation is exactly the same as that found in the preceding table, viz., $232.96. Or the amount may be determined in an other way, by adding to the quotient found by the sinking-fund method $172.96—the amount of inter est, at the same rate per cent, upon the original cost of the asset, which in this case is $60, namely, six per cent of $1,000.
Under the value-shrinkage theory, therefore, the sinking-fund and the annuity method are identical in their net effects. But the sinking-fund method deals only with the net shrinkage, whereas the annuity method represents the asset as both appreciating and depreciating. Thus, as regards the thousand-dollar machine, the annual depreciation would be represented as $232.96. The following form sets forth the entire account with this asset;. the supposition is that the depreciation allowance is credited directly to the ac count with the asset, instead of to a depreciation-re serve account.
Errors caused by rounding off decimals at the second place account for the discrepancy of five cents between this final balance and the estimated scrap value of $25. The entry to adjust at the end of the
first year would be as follows: Dr. Instrument account (interest) $ 60.00 Dr. Depreciation of instrument 232.96 Cr. Interest on invested capital $ 60.00 Cr. Instrument account or Cr. Reserve for depreciation of in strument 232.96 As the form above shows, the annuity method, con sidered in the light of the value-shrinkage principle, implies uniform efficiency of the instrument thruout its life. Also, according to this method, the total al lowance for depreciation during the life of the asset is considerably more than its cost. Needless to say, the annuity method has not status if it is judged by the allocated-cost principle, since that principle is not concerned in any way with changes in the value of the instrument after it has been purchased.
6. Level or constant depreciation method.—When the level or constant depreciation method is used, the depreciation assigned to one unit-period is the same as that assigned to any other unit-period, and the total assigned during the life of the instrument just equals its depreciable value, i.e., its cost minus its estimated scrap-value.
The formula for computing the amount of the periodic depreciation is D c R (VII) In this equation, D represents the amount of the depreciation, C the cost, R the scrap value, and n the number of periods constituting its estimated life. The periodic depreciation percentage, d, stated as a decimal fraction is given by the formula d D C—R C nXCThus, in the case of the $1,000 instrument, in the previous illustrations, since the estimated scrap value is $25 and the life is 5 years, D = $1,013° $25 -- $195, 5 and d = or .195 or 19.5 per cent.
7. Significance of the level-depreciation method.- Since the costs or present worths of the successive years' services, not counting those of the last year, are considered equal, according to the allocated-cost theory, it is evident that if the level-depreciation method were to be justified, the service-rendering ca pacity—the efficiency—of the machine would have to increase. But the service-rendering capacity of a productive instrument does not increase with age— quite the contrary—and therefore the allocated-cost principle does not theoretically warrant the level-de preciation method.