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will be credited and the particular asset purchased will be charged,
thus making no change in the total assets. If the allowance is in-
vested in the property, fair return presumably will be earned. If this is
not immediately possible, so that the allowance must be held for a time
until needed in the property, it should be invested in outside securi-
ties, and the deficiency of fair return, if any, should be charged next
year in operating expense, and the excess above fair return, if any,
should be credited to income, reducing the necessary payment of the
public on account of depreciation. Only on such an assumption is it
fair to use fair rate of return in computing annuities. If the com-
pany owner must take his chances of earning more or less than the
fair rate of return on the annuities, then that rate should be used
which experience shows to be probably obtainable. In the case of a
reasonably growing property, investment in the property will usually
be possible though perhaps not immediate; in the case of a stationary
property, it will in general be impossible.

There is no ambiguity in the foregoing described method; no
chance for confusion ; the method is exactly analogous to that described
under the straight-line theory of depreciation.

The Entries Under the Sinhing-Fund Method should be, and are
explained, as follows:

Operating income, in the amount of the annuity, is charged to
operating expense in order to keep the operating income account
correct. As fair return is based on book value, there must be no reduc-
tion in this below original value, except as units are retired, so there
will be no credit to depreciation reserve, which will reduce book value,
but to some other liability accoimt as trust fund reserve. Among the
assets, cash or other assets will be credited, and sinking fund charged
with the amount of the depreciation annuity. The fund will be
credited only as cash is expended for the replacement of, or in lieu of,
units covered by the fund. The fund must be carried as an undis-
tributable asset belonging to and going with the property, to oifset the
trust fund reserve. Accounting must go on within the fund;
its cash must be invested and its income accounted for separately from
the operating income. If its cash is reinvested in the property itself,
purchasing new units and increasing the income producing but de-
preciating property, the earnings from such new units should be charged
to the fund. These units being depreciating units must also earn
depreciation annuities, which must in turn be invested and in effect
a new sinking fund must be set up for each addition to depreciating
property. It would be manifestly impossible to separate the earnings
of the several units of a complex property, but the operating income



VALUATION OF PUBLIC UTILITIES 1475

can be separated into two parts, respectively proportioned to the
investment from the fund and that from other capital, and its due
share charged to the fund. This will be but an approximation to the
truth and again it must be remembered that the additions are of
depreciating property. Moreover, in a valuation it will be impossible
to distinguish between units purchased from the fund and those
acquired otherwise. If the fund is invested outside the property,
no difficulty arises in determining the income. Theoretically, if
it is invested so that it earns less than the rate assumed in
determining the annuities (which, as has been pointed out, should
be the fair rate of return), the deficiency may be put in operating
expense and so collected from the public ; and, if it earns more than the
fair rate of return, the excess should be credited to the public sinking-
fund reserve to the relief of the public. However the fund is invested,
it must be held to be a part of the undistributable property, an asset
to be included in a valuation offsetting trust fund reserve.

The foregoing procedure is more complex than that involved in
the compound-interest method, and the sinking-fund method has no
advantage over that method in any way. It does not show directly in
the property account what the accrued depreciation is at any time, as
does the compound-interest method, but presents the fiction — possibly
justified by the existence of the trust fund reserve — of an undepre-
ciated property. It must be remembered that the company owner
has two elements of value in his property — money and time — that
is to say, cost and length of service life, both of which must be pro-
tected. It is not enough that when a property unit goes out of service
there shall be a new unit of equal cost to replace it, but the new unit
must have a service life equal to that of the retired unit, or, there
must be more imits of less life, or less units of more life. Under a
strict application of the sinking-fund method, when the fund is in-
vested in the property, the foregoing result is realized only if the
funds be set up for each separate addition, purchased with fund
money, based upon the cost and service life of the addition. The
intricacies of this method when strictly followed, the confusion likely
to result when the fund is invested in additions to the property, and
the desirability of such (latter) form of investment, seem to make
this method of accounting, for depreciation of a growing property,
undesirable.

The sinking-fund methods in common use in the past have varied
from the foregoing procedure. Either annuity or full amount of
accruing depreciation has been charged to operating expense, and
credited to reserve, thus reducing book value. When only the annuity
has been charged and credited, the property account shows deprecia-
tion, but not enough, and explanation must be forthcoming to justify
the accounts and the distribution of public payments. The accounts



1-176 VALUATION OF PUBLIC UTILITIES

do not tell the whole story, as in the compound-interest method. If
the full accruing depreciation is charged and credited, no sinking
fund is required, and no explanation can justify any other distribution
of the public payments than that of the compound-interest method
which this procedure then becomes.

The sinking-fund methods heretofore generally used are incomplete
accounting methods, under the compound-interest theory, in that out-
side explanation is required to justify the accounts and the distribution
of public payments for return and depreciation. Since, as has been said
before, there is full and complete payment by the public for return and
depreciation under the compound-interest method, and since, with equal
rates for return and annuity, the payments are the same in total amount
under the sinking-fund method, they must be complete under both
methods, and it would appear that the sinking-fund method so widely
used must have been the outgrowth of a failure to comprehend the
problem completely. At any rate, it seems no longer desirable.
Comparison of Four Methods. —

(a) — Compared to Bond Payments. — The four methods of account-
ing for depreciation may be compared to four ways of making and pay-
ing bonds. Thus, the replacement method corresponds to a bond pay-
able n years from date, with interest at the rate of r per cent, per
annum until paid; the straight-line method corresponds to a set of n
serial bonds of equal amounts, with interest at the rate of r per cent,
per annum, giving diminishing annual payments; the compound-
interest method corresponds to a set of n serial bonds of unequal
amounts, respectively equivalent to the annuity accumulations of
the several years that the bonds run, giving equal annual payments.
The sinlving-fund method corresponds to bonds payable n years after
date, and requiring a trust fund to be accumulated with a bank or
trust company to meet the payments when due, also giving equal
annual payments.

(6)— On a Basis of Costs. — To compare the four methods thus far
described on a basis of cost, it will be assumed that the property is
made up of a single unit, costing $100, having no salvage value, and
lasting 20 years — of course, a purely hypothetical case. If interest
for all purposes is at 6%, the annual and total payments by the public
to the owners will be as in Table 3.

In preparing Table 3, the return imder the replacement and
sinking-fund methods has been figured as 5% on the undiminished
cost of the property unit, while under the straight-line and compound-
interest methods it has been based on the depreciated value of the unit.

The headings in Columns 9 and 10, while representing what is fre-
quently considered to be "depreciation" and "return", are not expres-
sive of the facts. The figures in Column 9 represent the sinking-fund
annuitv for each year, which is only a part of the depreciation, as may



VALUATION OF PUBLIC UTILITIES



1477



be seen by reference to the total at the foot of the column, which is
only $60.40 instead of $100. The remainder of the depreciation is
incorporated in the figures in the column headed "Return", where in
each year it represents 5% on the difference between the undiminished
cost of the unit and its depreciated value in that year. Of the $100
represented by the total of this column, $39.60 is "depreciation" and
$60.40 is "return".

TABLE 3. — Comparison of Combined Annual Depreciation and Re-
turn Upon a Property Unit' of 20-Year Life, Under Different
Methods of Providing for Depreciation. Assumed Rate 5 Per
Cent.



^


Replace-


Straight Line


Compound-Interest


SiNKING-F


U.N'D


o

•3


Method.




Method




Method.




Method.


4-


d


i ri


d




i d


d




i> d


d












es












OS




03


as


0)


o


^ 3




S
g


©Is
P-5




o


(1)


(2)


(3)


(4)


(5)


(6)


(7)


(8)


(9)


(lO)


(III


1


$5.00


$5.00


$5.00


$10.00


$3.02


$5.00


$8.02


$3.02


$5.00


$8.02


2


5.00


5.00


4.75


9.75


3.18


4.84


8.02


3.02


5.00


8.02


3


5.00


5.00


4.50


9.50


3.33


4.69


8.02


3.02


5.00


8.02


4


5.00


5.00


4.25


9.25


3..'i0


4.52


8.02


3.02


5.00


8.02


5


5.00


5.00


4.00


9.00


3.68


4.34


8.02


3.02


5.00


8.02


6


5.00


5.00


3.75


8.75


3.86


4.16


8.02


3.02


5.00


8.02


7


5.00


5.00


3.50


8.50


4.05


3.97


8.02


3.02


5.00


8.02


8


5.00


5.00


3.25


8.25


4.26


3.76


8.02


3.02


5.00


8.02


9


5.00


5.00


3.00


8.00


4.47


3.55


8.02


3.02


5.00


8.02


10


5.00


5.00


2.75


7.75


4.69


3.33


8.02


3.02


5.00


8.02


11


5.00


5.00


2.50


7.. 50


4.93


3.09


8.02


3.02


5.00


8.02


12


5.00


5.00


2.25


7.25


5.17


2.85


8.02


3.02


5.00


8.02


13


5.00


5.00


2.00


7.00


5.43


2.59


8.02


3.02


5.00


8.02


14


5.00


5.00


1.75


6.75


5.70


2.32


8.02


3.02


5.00


8.02


15


5.00


5.00


1.50


6.50


5.99


2.03


8.02


3.02


5.00


8.02


16


5.00


5.00


1.25


6.25


6.29


1.73


8.02


3.02


5.00


8.02


17


5.00


5.00


1.00


6.00


6.60


1.42


8.02


3.02


5.00


8.02


18


5.00


5.00


0.75


5.75


6.93


1.09


8.02


3.02


5.00


8.02


19


5.00


5.00


0.50


5.50


7.28


0.74


8.02


3.02


5.00


8.02


20


5.00


5.00


0.25


5.25


7.64


0.38


8.02


3.02


5.00


8.02


Depreci


ation 100.00




















Total.


$200.00


$100.00


$52.50


$152.50


$100.00


$60.40 $160.40


$60.40


$100.00


$160.40



Although the total payments by the public for return and depreci-
ation appear from the footings, under the various headings, to be
different under the different theories, this apparent variation is the
result of ignoring interest on the payments or, in other words, the
effect of time.

The real total cost of the unit in the 20 years, for interest
and depreciation, is the accumulation of the annual allowances for
these items at compound interest during the 20 years, and this is
the same for the four methods and is equivalent to the sum to which



1478 VALUATION OF PUBLIC UTILITIES

the cost of the unit would amount at compound interest in the life
time of the unit, as will be shown.

Referring to Table 3 : by the replacement method, the total return
of $5.00 each year may be considered as an ramuity which at 5% in
20 years amounts to $105.33, to which must be added $100.00, the
retiring payment, making $265.33. By the straight-line method, the
payment for the first year is $10.00, which at compound interest for
19 years amounts to $14.57; the second payment is $9.75, which in 18
years amounts to $13.92 ; and so on, the total for the 20 years summing
$265.33, as before. The payments by the two other methods are alike,
$8.02, and these, treated as annuities, also amount to $265.33 in 20
years; but $100.00, the first cost of the unit, if placed at compound
interest at 5% for 20 years, also amounts to $265.33. Thus is the
statement of equal cost for the four methods and its equivalent shown
for the particular case. It may be proved mathematically, as follows:

The sum to which any principal, P, will amount at r compound
interest in n years is

*S = P(l + r)" (.3)

The annuity F that must be set aside each year at r compound

interest in order to realize P dollars in n years is

P r
F = (4)

^ (1 + r)"— 1 ^ ^

and conversely the sum, P, to which an annual allowance at r com-
pound interest will amount in n years is

p^Fm + rT-n ^.j

r

The total cost of any unit in n years, being the amount of the total
allowances for interest and depreciation at compound interest, if n be
taken as the life of a unit costing P dollars, then:

By the replacement method, the annual payments are Pr for in-
terest, and, at the end of the nth year, a gross sum of P dollars to
make good the worn-out item. In the n years, the total cost will be
the sum of the annual allowances of Pr dollars at compound interest
plus P dollars. Therefore, using Equation (5), in which F becomes
Pr, we have

Total cost = 1^ — ^ ^ + P = P (1 + r^. ... (6)

r

which by Eqviation (3) is the amount of P at compound interest for

n years.

By the straight-line method, the annual contributions are not equal.

P
They are for each year — for depreciation, and a var3ing amount



VALUATION OP PUBLIC UTILITIES 1479

for interest, as the depreciation is deducted from the principal. The
series of payments for interest may be expressed as follows:

First year P r



r



Second year ( P ) r

Third year ( -^ ~ ' )^

etc.,
nth year (p - ^" " ^^ -^ )

The total annual contribution then will be

P
First j'ear 1- P r

n

Second year + (p ) r

P / 2 P

Third year + ( P ^ r

n \ n J

etc.,

nth year \- ( P — ^^ ^ ) r

n \ n J

Substituting those separate sums in a series of equations based on
Equation (3) making the exponent of the parenthesis n- — 1 for the
first equation, n — 2 for the second, and n — n = for the last equa-
tion, there results :

For the first-year allowance .. S = ( V Pr^ (1 + rf



\n — 1



For the second-year allowance. >S = ^ (^ ) '1 (-'■ + 'O""^

tP / 2 P \ ~i
h (P J r (l + r)"-3

etc..

For the nth-year allowance S = — + ( P — ) r

n \ n /

and the sum of these several sums is P (1 -f- ^)"j the same as given by
Equation (3).

By the sinking-fund method, the annuity payment. Equation (4), is

Pr

-, and the total annual payment is this plus interest, Pr.

(1 + r)"— 1' ^ ^ ^



1480 VALUATION" OF PUBLIC UTILITIES

Hence the total cost in n years is the accumulation of these two items

Pr
or, substituting, — + Pr for the i^ of Equation (5),

Total cost = Pr ^ '^ ^ =-= =- = P (1 + rf..{7)

r

or the same as by the replacement method, and as Equation (3).

Since the compoiuid-interest method involves exactly the same
annual contributions as the sinking-fund method, the total cost must
be the same, or, P (l-j-r)'*.

For the particular example assumed, namely, a $100.00 unit lasting
20 years, with interest at 5%, the total sacrifice for the 20 years, by
any one of the methods, is

S = P (1 + r)" = $100.00 (1.05)20 ^ $265.33.

(c) — With Eespect to Legality and Safety. — The four methods
showing identical costs, any one of them seeming to be most convenient
may be chosen for a particular property or for a part of a property,
provided only that it is legal, safe and fair, but the sinking-fund method,
being incomplete within itself, is not recommended. The replacement
method is open to the danger that a Court, valuing a property at any
time, will charge against it whatever depreciation may be found, hold-
ing that depreciation of physical property is always loss of value of in-
vestment, and that, under the Knoxville decision, it must be deducted,
whether or not, under accounting methods theretofore used, the depre-
ciation had been collected from the public. Thus, the property would
lose the accrued obligation of the public to pay for depreciation, which
obligation must be considered an asset of the property in an equitable
treatment of property maintained under the replaceinent method of
accounting. When the method is ordered or authorized by a public
regulating body, as has been done in the past by the Interstate Com-
merce Commission with respect to all railroad property, and is still
approved for most classes of such property, it should be considered
to be legal and should be safe, but not always has it been safe.

The straight-line and compound-interest methods plainly are
safe, as they return the full amount of the depreciation presumably
as it accrues.

The sinking-fund method perhaps requires interpretation to make
clear when it is safe, as the public apparently does not return
the full amount of the depreciation to the owner, and unless fully
understood and properly applied by regulating or valuing bodies, it is
not safe. Since the total aimual payments for interest and depreciation
by this method are exactly the same as by the compound-interest method,
it must follow that if full payment is made in one case it is also made
in the other, and that it is merely the bookkeeping distribution of these



VALUATION OF PUBLIC UTILITIES 1481

payments which makes it appear that full payment is made in one case
and not in the other. Under the sinking-fund method of nccounting,
the annuity assumed to be paid by the public, together with the interest
on the accumulated allowances, must be charged each year to the fund
held in trust for the property. The fund must be treated as a trust
fund belonging to the property, or due from the public, and equivalent
to physical property of equal value.

If for the purpose of public purchase or capitalization a valua-
tion is made at any time of a property maintained under this method,
depreciation will exist in amount equivalent to the accumulation of
depreciation annuities and interest, if the various original estimates
on which the annuities were figured have agreed with the facts as they
have developed in the life of the property. To the depreciated value
found will be added the accumulated allowances not invested in the
property but treated as an asset due from the public and neither the
company owner nor the public will lose anything, since 100% of
original value will be found and the accounting may go on as before.
If part of the fund is invested in additions and betterments, it will
be difficult, if not impossible, to separate these from the original prop-
erty, and they will be included ordinarily in the valuation; but the
fund accounts should show what sums have been thus invested and
what amounts remain in the fund as other assets. Only the portion
of the fund not invested in the property can be added to the depre-
ciated value. Again 100% of original value will be found to result,
and purchase price will be so fixed, or the accounting may proceed as
theretofore. The sinking-fund method, being likely to cause confu-
sion and possible injustice, should not be used in depreciation account-
ing, but, if used, difficulties may be somewhat avoided if the fund is
kept invested in assets wholly distinct from the property.

If a valuation is for the determination of fair return for the future,
either of two courses is open: (1) The property may be listed at full
value for the determination of interest, and full value and total service
life for the determination of depreciation annuity; or, (2) The prop-
erty may be listed at depreciated value for the determination of in-
terest, and depreciated value and remaining service life for deprecia-
tion annuity. Either method is equitable, and both give the same
total annual allowance. Thus, referring to Table 2, if a 20-year item
costing $100.00 new has been in service 15 years, and interest is figured
at 5%, the theoretical remaining value, under the compound-interest
theory, is $34.74. The interest for the succeeding year is $1.73, and the
depreciation annuity to retire the item in 5 years is $6.29, or a total
of $8.02, which is just what is required to cover interest and annuity
for the item considered new. A danger here — shown- by experience to
be real — is that the property will be listed at depreciated value for
interest and full value and total service life for depreciation annuity.



1482 VALUATION OF PUBLIC UTILITIES

Rate on Depreciation Allowances. — As having some bearing on
accounting methods, the disposition of allowances, and the rate to be
used for these allowances, the Committee desires to emphasize again
the fact that the entire reasonable sum devoted to public service is
entitled to a uniform fair return. While thus devoted, and while, as
a matter of convenience or to conform to law, allowances for deprecia-
tion may be returned to the company owner approximately as the
depreciation goes on, thus theoretically reducing the capital in the
service, yet the obligation of the owner to keep the investment in the
property intact, makes it improper for him to consider these allowances
other than as a part of the property not to be distributed as income, or
returned to his pocket, without corresponding reduction of capital, if
not needed for the maintenance of the property. Therefore, he is enti-
tled to a full fair return on the undiminished investment so long as its
value is properly maintained in accordance with law. Hence the allow-
ances returned to the investor to cover accruing depreciations should
earn the same fair return as the value remaining in the property. To
accomplish this these allowances are best invested in the property, but,
when this is not possible, the amounts not thus invested should be
placed in a fund sacred to the property, and the earnings of this fund
should be accounted for — if less than a fair rate of return, the defi-
ciency should be charged in operating expense; if more, the excess
should be credited to income. Thus only will the public pay its just
payment and the property receive its fair return and no more. This
may be difficult or impracticable of exact accomplishment.

This means that interest on depreciation allowances should be at
the same rate as the fair rate of return, and that this is the rate to
use in computing annuities, and allowances under the compound-
interest theory.

Accounting Depreciation, the Result. — As a result of approved
accounting methods that include a reserve account and that have been
used from the beginning, there will appear on the books at any
given time a sum that may be called accounting depreciation. For
those items maintained by the replacement method and for which
reserves are not set up the sum will be zero, there is no accounting
depreciation under this method. The Interstate Commerce Commis-
sion has recommended that reserves be established for all properties,
and there is much to be said in favor of this recommendation. If
a reserve account is set up, the balance in that account shows the
amount of accounting depreciation for property in use.

If no reserve account is set up, there is no account that will
show at once the amount of the accounting depreciation. The way to
determine it is to compare the presently carried property statement
with the original costs of the property in use, if these can be found;



VALUATION OF PUBLIC UTILITIES 1483

the difference is the accounting depreciation for the methods of book-
keeping in use. Under the sinking-fund method, there should be no
reserve set up against the property, but the balance in the trust



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