**DEPRECIATION.**

**(CONCLUDED.)**

A good, well-built, substantial reservoir is like good wine—is often the better for age. It is said that the Pool of Hezekiah of Jerusalem is still used for the purpose for which it was originally intended, having had a life of nearly 2,600 years. It is hard to see where a substantially built reservoir can have much depreciation so long as it is useful at all. Generally it has its life terminated either by being of too small capacity for the growing population; or by reason of the increasing value of its site for other purposes or by new requirements for additional pressure. Climate has much to do with deterioration in reservoirs; the severe Northern winters, with their attendant ice, strong winds, alternate freezing and thawing, are, perhaps, the most severe conditions which a well-built reservoir must endure, while the dry climate of Egypt, Greece, and Italy is particularly favorable to structures of this class, as is evidenced by the numerous examples left from antiquity. Aside from the action of the elements, rapid filling and emptying, due to insufficient capacity, is a common cause of deterioration. Of poorly built reservoirs, one can only say that they lead an uncertain and precarious existence, and their duration of life may, perhaps, be the subject of an intelligent guess. Thus far it is only possible to mention a few facts out of many which might be cited, covering only three or four salient factors in waterworks plants. One general law remains to be noted.

Whatever may be the special fluctuations in the life of the individual factors of a waterworks plant, such as have been hereinbefore described, it is evident that, as a general truth, some average rate of depreciation holds good for all of the structures of the united plant, or for any given number of structures of the same kind. A good many estimators have, without much thought, assumed that this average rate of depreciation was uniform, and, to estimate present value, they assume the life of the structure, take its age at the date of valuation, and figure the decline uniformly for the given period. I bis method may be called the straight-line method of fixing depreciation, as shown on some of the accompanying diagrams. Now. it has been pointed out several times in this paper that scarcely any part of a waterworks system depreciates uniformly from the beginning to the end of its life. In nearly every case, depreciation is slight in the early years, when materials are new, and usefulness is at a maximum, while at the latter end of life loss of value is rapid, and usefulness declines very fast. This is a point to which it is only necessary to call a waterworks man’s attention to in order to receive his assent, because it accords with his experience in such matters Nearly every machine or structure is worth a good deal until the day it is taken out, and even then has some other field or usefulness, or at least a scrap-heap value. And it has been pointed out that, in some cases, at least, the value of a machine or appliance is, perhaps, even greater a year or two after it has been installed than it was at its purchase. The advocates of a straight-line theory of depreciation have felt that it did not adequately represent their experience, especially in the early part of structural life, in that it made some very great deductions of values which were obviously unwarranted and sometimes unjust. In order to reconcile experience to rule, the users of straight-percentage depreciation have had recourse to the expedient of assuming an undue length of life for the structures of water plants which are not entirely in accordance with known facts, but, nevertheless, give a resulting loss of value for certain early periods in conformity with experience. In the appraisement of the Sheboygan water plant. Mr. Henczette Williams, C. E.. of Chicago, who was chairman of that board, first called the writer’s attention to this difficulty, and proposed that, instead of figuring loss of value on the usual straight-line percentage, with, perhaps, undue lengths of life, when it did not accord with experience, the proper period of life should he adopted, and the depreciation should be figured on the basis of a sinking fund set aside from year to year and compounded at very low rates of interest, such as would be obviously proper for such an investment. Such a sinking fund should at the close of the life of any structure produce a sum sufficient to renew that structure in its original condition. The adaptability of sinking fund methods to depreciation will be further seen, when it is shown that a sum set aside at low rates of interest and compounded increases very slowly at first, more rapidly later, and very rapidly at last. This is precisely what happens in the average case of depreciation. The loss of value is at first slight. Later on it becomes much more pronounced, and at the last is rapid. Such a rate is denoted by a convex curve on a diagram commencing and ending with the beginning and end of the assumed life. Any particular degree of convexity is obtained by using various rates of interest, so that average conditions observed by experience may be duplicated in theory without difficulty. Of course, the minor fluctuations in the life of any given structure arc not followed, but are subject to especial consideration. The rates of interest on sinking funds arc always low, for the reason that it is practically impossible to reinvest interest payments in such a way as to realize the theoretical effect of compounding. The sums to reinvest are usually small, and expenses and commissions reduce the net returns so that from two to three and one-half per cent., depending upon the amount of money to be handled, is all that can be safely counted upon. The sinking fund always seems entirely fair and just to any one to whom it has been suggested. It makes the impression of being the only proper and reasonable way to figure loss of value on ordinary structures. The computations are, however, somewhat complicated and difficult for the average busy man to remember or use. In order that the method may become more favorably known, a series of tables has been computed by Mr. D. W. Mead, of this society, which have been a great aid in a number of recent appraisements.

The following formula is subjoined:

The annual investment necessary to renew a plant at the end of a given number of years is obtained by the formula (4) deducted as follows:

*Let* “p” equal yearly investment.

“r” equal rate of interest.

“n” equal number of years.

“s” equal the amount at the end of “n” years. The amount “a” of “p” dollars at compound interest for “n” years is obtained by the following formula commonly used for computing compound interest.

a=p (1+r)^{n} (1) If “p” = $1.00, the formula becomes a= (1+r)n (2)

If $1.00 be invested each year for a term of years the sum of the amounts of each year’s investments compounded singly for the number of years that each runs is represented by the following formula in which “s” equals the total sum.

S=(1+r)+(1+r)^{2} (1+r)^{3}-(1+r) (3)

From which as the above is a geometrical series.

s=(1+r)n+1/r(1+r) (4)

To illustrate the use of the above formula, let us assume that we are to determine the depreciation on a pumping engine whose present cost of duplication is $15,000, that has been in service, say 18 years, and after carefully considering all the surrounding circumstances, we fix upon a probable life of thirty years. Let us compute the present worth of this engine. We will assume two and one-half per cent, as the proper rate of interest to be used in the computation. From formula (4) we obtain the amount of $1.00 invested annually for a period of thirty years, which equals $45.00. Therefore, to duplicate an engine costing $15,000 at the end of its life, we must lay aside each year as many dollars as 45 is contained in 15,000 or $333.33. We next obtain by the same formula (4) the amount of an annual investment of $1.00 for a term of eighteen years at the same rate of interest; this equals $22.94 and the amount, therefore, of $333.33 invested annually is $333-33 x $22.94 centals $7,646.59. which is the depreciation in the engine at the present time.