obtained by dividing the sum of the wearing values of all such portions of the plant by the number of years of their life. Or, in the case of a large plant with elements each of different life, the mean life is found by dividing the sum of such quotients into the sum of the wearing values of all plant elements, i.e., the wearing value of the entire plant. This can be illustrated best by a concrete case. Let the wearing value of all plant elements that can be useful for 5 years be $50,000; for 10 years be $100,000; for 15 years, $75,000; for 20 years be $100,000. Then, $50,000 ÷ 5 = = $10,000 This shows that the annual reserves should be $30,000, which would be the same as if it had been said that the entire plant, having a wearing value of $325,000, had a mean life of approximately 10.8 years (325,000÷30,000). 105. Danger of misconception of mean life. In the preceding section an example is given illustrating the correct method of determining the mean life of a property. When thus determined the mean life becomes a figure by the use of which the property of a utility in one place may be compared with a similar property in another place. Naturally local conditions may have an important effect in causing a variation, but, generally speaking, it is possible to say that the mean life of a water plant is x years, of a gas plant y years, or of a telephone plant z years. But when we speak of the mean life of such a property as was considered in the preceding example as 10.8 years, it is not meant that $30,000 is set aside each year for 10.8 years and that the renewals required during that period are not made when required. The mean life is simply a practical figure derived by a careful study of each unit and is to be used in determining the annual reserve for the plant as a whole. The reserve for each particular unit is made each year and is held for that unit in the funds until its life has been exhausted, at which time the wearing value of that unit is taken from the reserve to pay the cost of renewal of that particular unit. Possibly the idea of mean life will be clearer if the following thought is kept in mind. Take the case used in the preceding section and suppose that the plant does not grow and that the plant cost does not change, then, so long as the utility is in existence, using units which have been worn out and replaced time and again, there will have to be set aside as a yearly reserve for depreciation $30,000. Mean life is, therefore, simply the number of years which will elapse before an amount equal to the total cost of the perishable property will have been paid from revenue for reserves for renewals. The mean life is not a life dating from the original construction of the plant. It is true of a given plant at any time, now or in the future, provided the plant does not change in character or in cost. The failure to appreciate that the reserves for each unit accumulate for the benefit of that unit throughout that unit's life, and that mean life is simply a composite figure useful for determining what proportion of the total cost of the plant shall be set aside as the annual reserve for the entire perishable property, is the cause of most of the existing misunderstanding relative to the sinking fund method of accumulating reserves for renewals and the cause of many of the arbitrary rulings in favor of the straight line method.1 106. Adoption of "sinking fund" or "straight line" 1 See Manhattan Railway Co. v. Woodbury, 203 N. Y. 231, (Oct. 17, method of making reserves in practical cases. The discussion in the preceding sections of this chapter has been very largely theoretical and has been intended to present the fundamental principles involved in maintaining the capital value of the undertaking. When the practical case is presented to the management of a public utility as to which basis shall be adopted in making reserves for depreciation, the question must be answered very largely by a decision as to whether the reserves for depreciation are to be invested in extensions to the plant or not. If the reserves are to be invested in plant enlargements, then the straight line method must be employed. On the other hand, if the reserves for depreciation are invested in outside securities, then, as such funds have the benefit of interest accumulations, the use of a straight line method would result in abnormally large sums being set aside, and, in consequence, the sinking fund method is to be preferred. 107. Straight line method.-The reason why the straight line method should be employed, when the reserves for renewals are invested in plant, is apparent when the operation of the sinking fund method is considered. This can be understood most readily by a consideration of a practical example. Let it be assumed that the plant of an undertaking had cost $100,000 (assuming zero salvage value) and that it had a mean life of 10 years. Let it also be assumed that the undertaking operating the plant is entitled to a return of eight per cent. Then, during each of the ten years, the gross income must be sufficient to pay taxes, to pay the operating expenses, to pay eight per cent on the investment of $100,000.00 and to pay to a reserve fund such an amount as will aggregate at the end of ten years the original investment. For the present example, let it be assumed that each increment to this fund has been invested each year in new plant needed and required by the public. If the sinking fund method is employed in figuring reserves for renewals, the undertaking must pay not only the annuity but the interest on the sinking fund. If the straight line method is employed, the increments to the reserve fund will be the original cost divided by the years of life, or $10,000. The operation of this fund for each year of life by each of the above methods of accumulation is shown in the following table: From this it is seen that, by the sinking fund method, the sums of the payments to the reserves, for annuity and interest, during the early years of the life of the plant are less than they would be if the straight line method was used, but, later on, the annual payments to the sinking fund become very much larger, in the 10th year $13,799 and $10,000 respectively. The results of the two methods, however, are the same, in that, at the end of the life of the plant, there has been accumulated $100,000 by both methods. In other words, there has been contributed by the public for the purpose of reserves $100,000 by either method during the ten years that the plant has been useful. When it is remembered that the plant is made up of a large number of units, which have been brought into service at different dates, and is composed of units of different lives, it will be appreciated that the sum of the increments for the reserves for each unit would be not the amounts shown as the yearly total of the sinking fund column but rather figures approaching those shown in the straight line column. But even supposing that we were dealing only with a single plant unit; as the $100,000, the entire original cost, must be recovered by the undertaking from the public, clearly it is simpler and more rational to make the payments uniform for each year than to set up an artificial method such as that of the sinking fund, wherein different amounts must be set aside each year. It is true that, in some cases, it is desirable to make the burden upon the public as light in the early years of the operation of an undertaking as possible, but, for most public utilities, such an argument has no force. Moreover, the inequality of the demands for depreciation reserves each year would indicate, at least theoretically, a necessity for an increasing or variable charge for the service or utility furnished. For the above reasons there seems to be no question but that, when reserves are used for needed and useful extensions of plant, the straight line method of making reserves for depreciation should be employed. 108. Sinking fund method. For practical reasons there seems to be no doubt as to the desirability of making uniform increments each year to the reserves for depreciation. By "uniform" it is not meant that the |