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straight-line theory when applied to newly created properties, because
it requires the setting aside of a less sum to offset accruing depreciation
in the earlier years of the enterprise, when expense is likely to be rela-
tively high and income relatively low. On the other hand, when applied
to a unit for which the cost for repairs and upkeep is much greater
toward the end of its life than at the beginning, this theory bunches
tlie larger deductions from income for repairs and depreciation; and
the straight-line theory would tend to produce less inequality in the
deductions throughout the years of service life, so far as this influence
is concerned.

With some units, such as locomotives, the expense for repairs is
so large that the depreciation deduction is relatively insignificant and,
so far as inequality of deductions is concerned, it is of less consequence
which theory is adopted than it is when units like water pipe are
under consideration, because the item of repairs on such iinits is
relatively insignificant when compared with the accruing depreciation.

The discrepancy between accrued depreciation under the straight-
line theory and under the compound-interest theory is much greater
when the omits are long-lived than when they are short-lived. Thus,
a 10-year unit having no salvage value loses half its value in 5 years
under the straight-line theory, regardless of the rate of interest, and at
5% under the compound-interest theory it loses nearly as much —
about 44% of its value in the same time (45.1% at 4% and 41.6% at
7%) ; but a 50-year unit, losing half of its value in 25 years, under
the straight-line theory, loses only 22.8% of its value at 5% under the



VALUATION OF PUBLIC UTILITIES 1465

compound-interest theory. It will be shown later, however, that
the total cost for service during the service life of a property unit,
including interest on the value in the unit, depreciation annuity or
allowance, and interest on both these annual costs, is exactly the same
under one theory as under the other, and indeed is the same as under
the replacement method of maintaining property already described,
but, notwithstanding this equality, the theories are not equally applic-
able to all classes of property, as will also be shown later.

The Unit-Cost Theory. — This theory holds that the depreciation
of a service imit at any stage of the service life is the difference between
what it cost new and what could be paid for it in its depreciated con-
dition without increasing or diminishing the cost per unit of service
during its remaining service life. In other words, it is equivalent to
the difference in worth of two units which would perform the same
service at the same total cost per imit of service, one having an
estimated service life equal to the total estimated service life of the
unit under consideration, and the other an estimated service life equal
to the estimated remaining service life of the unit. Comparing the
relative economy of two such units in general, the engineer solves a
question like this : If Unit A costs $100 and lasts 20 years, how much
can be paid with equal economy for Unit B for the same service, if
it will last but 10 years? The costs must include all costs necessary
to put the units in place and at work. The estimate should include
consideration of the effect of the lessened output of service and greater
cost for operation and repair of the units, as they draw near the ends
of their lives; but this effect is sometimes ignored. When excluded,
the "unit-cost theory" becomes the "compound-interest theory", which,
therefore, may be considered to' be a special case of the "unit-cost theory".
On the basis of this theory of difference in worth, considering Unit A
as the unit new and Unit B as the depreciated unit, a formula for
the present worth of the depreciated unit is developed thus, ignoring
output and cost of operation and repairs: The two units are com-
iiared on the basis of equal annual cost for interest and depreciation
annuity. If C be the cost new of Unit A and c the present worth of
the depreciated unit, B; r the rate of interest; N the estimated service
life of A ; and n the estimated service life of B ; the total annual costs
for interest and annuity will be :



For A. . . , Interest, C r, .... Annuity,



C



(1 -f r)^_ 1

c r
For B . . . . Interest, c r, .... Annuity, ;,

" (1 + r)"— 1

Making these two equal and solving for c gives

r(l + r)"— 11



146G VALUATION OF PUBLIC UTILITIES

and C — c would be the depreciation during N — n years, which will
be found to be identical with the 8 of Equation (2), if {N — n) be
substituted for n in that equation. This is the compound-interest
theory.

If the items of increased cost of operation and repairs be included,
the application of the unit-cost theory becomes quite complex. The
comparison is made in the same way by making total annual costs
equivalent, but the annual cost is first reduced to cost per unit of
service by dividing by the estimated output capacity of the new and
depreciated property units. Thus, if A and a are costs of operation
and repairs for the longer- and shorter-lived units (new and depreciated
units), respectively; C and c the respective costs, G being the actual
or estimated cost of the new unit of longer life and c the price that
could be paid for the shorter-lived (depreciated) unit to produce the
same cost per unit of service; B and & the respective depreciation
annuity multipliers of C and c under the sinking-fund theory; if D
units of service may be realized annually from the longer-lived (new)
unit and d units from the shorter-lived (depreciated) unit; and if r
be the rate of interest, the following equations of annual costs may be
written :

A + B C + r G _ a + h c + r c
lb ~ d

Solving this equation for c, the price that could be paid for the
shorter-lived (depreciated) unit is found to be:

^{A + BG+vG) — a



The difference between G and c is the accrued depreciation at the date
of estimate. The amounts of B and h may be obtained from

B

and h



(1 + ry_ 1

r
(1 + r)" — 1



or from amortization tables. The capacities, D and d, of the new
and old units should be their actual or estimated capacities for service
under normal conditions, not the respective outputs under varying
business conditions which might readily make the older unit more
valuable than the newer one.

The compound-interest theory produces, in its application in
accounting, equal annual charges for the sum of interest and de-
preciation allowances, while the unit-cost theory is based upon equal



VALUATION OF PUBLIC UTILITIES 1467

annual charges for the sum of interest, depreciation allowance, and
operating expense (including repairs) per unit of output. This
theory is probably the soundest theory of depreciation, and, when
applied with intelligence, probably furnishes the truest measiire
of accrued depreciation. It is not readily applicable to accounting
purposes, because its formulas cannot be tabulated, but it may be
used for the purpose of checking the results of the application of
other theories and for finding the accrued depreciation of any property
at any time, in connection with a public valuation, when there has
been no proper accounting in the past, or for finding remaining value
for the purpose of inaugurating a proper accounting system for an
old property.

Methods of Accounting Applicable to the Straight-Line and Compound-
Interest Theories of Depreciation.

This discussion will be approached from the standpoint of the
proper payments of the public to company owners and the proper
disposition of these payments by the company owners, rather than
from the standpoint of the bookkeeping methods of the company
owners. It should be remembered here that it is the duty of the
public served to pay operating expense incurred in its service, interest
on capital invested for its service, and the cost of property con-
sumed in its service; but operating expense is omitted from this
discussion as being independent of the other two items, which are
to some extent mutually dependent on each other, and because it is
only depreciation accounting that is now under consideration.

Accounting Under the Straight-Line Theory. — Under this theory
the public will pay the full amount of the depreciation as it accrues,
a certain uniform fraction of the whole cost, less salvage value, of any
unit each year of its life, and, paying this each year, thus returning-
a part of the invested capital, it should pay a correspondingly decreased
interest on invested capital each succeeding year. A tabular statement
of a particular example will illustrate: A certain vmit costs $100, has
a life of 20 years, and no salvage value. The cost or book value in the
unit and the payments of the public with interest at 5% are shown in
Table 1.

Remembering that the property must be considered to be a con-
tinuing property and that the company owner is bound to preserve the
integrity of the investment, it must be clear that the allowances paid
by the rate-payers belong to the property and must be considered to
be a part of it. If taken out of the property, the investment is
impaired, and if paid to the stockholders, then, although they lose
nothing, the bondholders' security is reduced, and the public and later
stock purchasers are not protected against the gradual destruction



1468



VALUATION OF PUBLIC UTILITIKS



of the property that should be a continuing property. Practically,
therefore, the depreciation allowances paid by the rate-payers must
be considered a part of the property, not to be distributed to the
stockholders, but to be kept in the property in some form, in order
to preserve the integrity of the investment.

TABLE 1. — Illustration of the Straight-Line Method.

Showing Book Value and Payments by the Public on Account of
Fair Return and Depreciation.

Property Unit Costing $100.00, Lasting 20 Years, No Salvage Value,
Interest at 5 Per Cent.



(I)


(2)


(3)


(4)


(5)
Combined


Age.


Kemaining '
value at end
of year.


Depreciation
allowance.


Return on

remaining value.

5%.


depreciation
and return,

5%.





$100.00


$5.00


$5.00


$10.00


1


- 95.00


5.00


4.75


9.75


2


90.00


5.00


4.50


9.50


3


85.00


5.00


4.25


9.25


4


80.00


5.00


4.00


9.00


.5


75.00


5.00


3.75


8.75


6


70.00


5.00


3.50


8.50


7


65.00


5.0O


3.25


8 25


8


60.00


5.00


3.00


8.00


9


55.00


5.00


2.75


7.75


10


50.00


5.00


2.50


7.60


11


45.00


5.00


2.25


7.25


12


40.00


5.00


2.00


7.00


13


35.00


5.00


1.75


6.75


14


30.00


5.00


1.50


6.50


15


2.i.00


5.00


1.25


6.25


16


20.00


5.00


1.00


6.00


17


15.00


5.00


0.75


5.75


18


10.00


5.00


0.50


5.50


19


5.00


5.00


0.25


5.25


20











■ Tot&ls.




\ $100.00


$52.50


$152.50


■^ — .


1 ...


-



VALUATION OF PUBLIC UTILITIES 1469

The allowances, being part of the property, ought to earn at the
same rate as the capital remaining in the property. That is, persons
having invested in a given property do not wish the investment
continually to vary either in amount or return rate, they desire that
the investment shall remain and shall earn at a uniform rate. The
allowances may be made to earn at the same rate as the remaining
property, if reinvested immediately in the property; they may earn
at an even greater rate, if fortunately invested otherwise; or they may
and probably will earn at a less rate, if placed in bank. Immediate
investment in the property itself, which should insure a continuance
of the full rate of return, may not be possible, often will not be when
a property is not rapidly growing, and lack of opportunity for other
investment may require the banking of allowances. It would seem,
therefore, that the public should make up the shortage in return, if
any there be. Theoretically then, proper accounting, under regulation
that determines fair return by which full return on the entire in-
vestment is secured, would charge to operating expense any deficiency
of return on allowances and credit to income any excess of return
over the fair rate.

When only a single item is considered, this method, as the table
shows, makes payments from year to year ununiform, with much larger
payments in the beginning than toward the end, which is undesirable
unless repairs are enough greater toward the end to equalize the pay-
ments. When there are many items of varying ages and expectations
of life, this defect largely vanishes. This is not a defect of the
accounting, but a necessary result of using the straight-line theory of
depreciation.

Accounting Under the Compound-Interest Theory. — Under this
theory, the public pays interest on the invested capital and depreciation
as it accrues, according to the compound-interest theory, but there are
different ways of considering these payments and distributing them in
the accounting, leading to two or more different methods of accounting.
The Committee will discuss two methods, namely: 1, the compound-
interest method, heretofore called the equal-annual-payment method,
and 2, the sinking-fund method.

1. The Compound-Interest Method. — Confining the discussion to a
single unit without salvage value, for simplicity, the amount of the
annual payment for interest and depreciation is found thus : If C be the
cost of the unit; iV" the years of service life; r the rate of interest for
fair return; and / the rate to be used in amortizing the property; the
total annual payment, P, by the public for fair return and depreciation
is given by

^ -^-^'^(1 + rr-i



1470 VALUATION OF PUBLIC UTILITIES

the second term being evaluated by logarithms or taken from amor-
tization tables. It will perhaps be plain that this should be the
payment for the first year, but for the second year the payment should
include interest on the amortization annuity of the first year, if
the public served is to pay the full depreciation as it accrues, and
the return should be on the original cost reduced by the first annuity,
just as in the straight-line method. For the third year, the cost,
or value, should be reduced by the depreciation payments of the first
two years, and the depreciation payment should include interest on
the sum of the two preceding depreciation payments, and so on.
Now it may be shown that if / equals r, all these several annual total
payments for interest and depreciation are identical in amount and
equal to that first above written for the first year. If r' does not
equal r, the total payments are not the same throughout the years.
Table 2 will show the remaining value, depreciation accruing, and total
payments on account of fair return and depreciation, when the amor-
tization interest rate is assumed the same as the fair rate of return —
5%, and when the latter is the larger — 7 per cent.

This method of disposing of the payments of the public on account
of fair return and depreciation was called, in the Progress Report,
the equal-annual-payment method, because it seemed to make the
total payments on account of operation, return, and depreciation more
nearly equal from year to year than other methods that had been dis-
cussed. As it does not necessarily do this in all cases, it might better
be distinguished from other methods by a more exact title. Under
this method the total payment of the public served, for interest and
depreciation, is divided into two parts, one the full annually accruing
depreciation under the compound-interest theory, and the other, return
on the depreciated cost of the unit. This is exactly analogous to the
straight-line accounting method, the only difference being in the amount
of the accruing depreciation. It is the complete and logical method of
accounting for the payments of the public for interest and depreciation
under the compound-interest theory of depreciation, and the method
will be given the name of the theory and will be called the "compound-
interest method", logically comparable with the use of the ''straight-
line method" under the "straight-line theory". Although the name
"equal-annual-payment" has gotten into valuation and accounting
literature, this literature is in a transition stage, and the Committee
thinks there is good reason in the preceding argument for the new
name.

2. The SinHng-Fund Method. — Sinking fund is a very old term and
has long been connected with the plan of providing for the payment
of a debt, which is to become due in the future, by setting aside and
investing a properly determined annuity, such that, with accumula-



VALUATION OF PUBLIC UTILITIES



1471



TABLE 2. — Illustration of Compound-Interest (Equal- Annual-
Payment) Method of Accounting for Depreciation.

Assumptions : Property having 20-year Life, Valued when Xew at $100.
Computations of Depreciation Allowances Based on 5% Interest,
Compounded Annually. Annual Return on Capital Invested, 5 and
7 per cent.



Age,
in


Remaining
value
at end


Deprecia-
tion during
year = de-


Return on Remaining
Value of Property at :


Combined Depreciation
and Return


years.

(1)


of year.

(2)


preciation
allowance.

(3)


5%
(4)


(5)


5%
(6)






1

2

3

4

.5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20


$100.00
96.88
93.80
90.47
86.97
83.29
79.43
75.38
71.12
66.65
61.96
57.03
51.88
46.43
40.73
34.74
28.45
21.85
14.92
7.64
0.00


.$3.02
3.18
3 33
3.50
3.68
3.86
4.05
4.26
4.47
4.69
4.93
5.17
5.43
5.70
5.99
6.29
6.60
6.93
7.28
7.64


15.00
4.84
4.69
4.52
4.34
4.16
3.97
3.76
3.55
3.33
3.09
2.85
•2.59
2.82
2 03
1 73
1.42
1.09
0.74
0.38


$.7.00
6.79
6.57
6.33
6.08
5.83
5.56
5.27
4.98
4.67
4.33
3.99
3.63
3.25
2.85
2.43
1.99
1.53
1.04
0.53


$8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8.02
8 02
8.02
8.02


$10.02
9.97
9.90
S.83
9.76
9.69
9.61
9.53
9.45
9.36
9.26
9.16
9.06
8.95
8.84
8.72
8.59
8,46
8.32
8.17






$100.00











tions of interest, the amount of the funds in hand when the debt
matures, will be equal to the amount of the debt. Only in compara-
tively recent years has the method been used for the amortization of
wasting properties. It is not so well adapted to this purpose, as to



1473 VALUATION OF PUBLIC UTILITIES

the amortization of a debt, particularly when the property is a public
utility. The method contemplates the creation and maintenance of
a fund which, considering the danger that it may be misapplied, is
against good public policy. Moreover, when a property is growing it
is good economy to use the annuities in extensions of the property ;
but there is danger, when this is done, that confusion in the accounting
will result, and there is always danger that some persons may confuse
annuities without their accretions, with the depreciation that the
annuities plus their accumulations of interest are intended to cover.
Under this method, the public served is credited with paying only the
annuity toward depreciation, its payments on this account, therefore,
being equal from year to year. The remainder of the total payment
for interest and depreciation is treated as fair return computed on the
undepreciated cost of the tmit concerned. As this is also always the
same, the total payments of the public for fair return and deprecia-
tion are always the same throughout the years, and the amount is that
above written.

Or'

P=Cr +

(l + r'>^-l

If r' equals r, the total annual payments agree in amount with those
of the compound-interest method, but, if the return rate is higher
than the amortization rate, as has been usual in practice heretofore,
the total annual payments by the sinking-fund method are larger in
the later years than those of the compound-interest method as shown
in Column (7) of Table 2, and thus the company owner is over-
paid. It has already been argued that the same rate of return
should be earned by all capital kept in or held for the property,
and hence it should be true that amortization rates of interest
should be the same as fair return rates, and they will be so considered
in the remaining discussion. Thus, the total annual payments by
both methods will be identical. As full payment for interest and
depreciation is made under the compound-interest method and the
same totals are paid under the sinking-fund method, full payment must
be made under this method.

To defend crediting the public with only the annuities toward
depreciation and full return on the undiminished cost of service units,
one of two arguments is invoked:

1- — The annuities are paid to the company before it has occasion
to use them to retire the units they are intended to cover; hence, the
company owner can earn interest on them, and therefore they are
discounted and considered parts of a fund held in trust by the company
owner and invested for the purpose of earning that interest necessary to
make it amount to the payment required of the public when the unit
covered is retired. As the company may not earn distributable profits



VALUATION OF PUBLIC UTILITIES 1473

Oil these annuities, it must be allowed its fair return on the undimin-
ished cost of the units covered until such time as the fund matures,
which in theory should be when the item is retired. If the fund is then
immediately reinvested in the property, the cost value goes on as
before; if, for any reason, it is withdrawn from the property, the cost
value should be reduced accordingly.

2. — The public is assumed to make full payment on account of
depreciation yearly as it accrues; but, because it makes these payments
in advance of the time when they must be used to retire the unit they
cover, the company owner will be able to use them to earn interest,
and hence the public charges the company for the use of these pay-
ments assumed to be due only when the item is retired; that is, the
public pays the present worth of payments assumed to be due on the
retirement of the unit covered. But, again, the company owner must
invest these annuities so that with their earnings they shall equal the
cost of the unit covered when it is retired; they cannot be used to earn
distributable profits, and, therefore, as before, the public should pay
a fair return on the undiminished coat value of the units concerned
until they are retired or the payments mature. Under this concep-
tion, it must be assumed, either that the last payment due is made
first and discounted, or that the growth of depreciation is reversed,
the larger annual growth being at the beginning and the smaller at
the end of service life. The conception seems not to be defensible.

Comparisons. — The entries to be made in the accounting under
the compound-interest and sinking-fund methods must of necessity
be different. Under the compound-interest method, the book value
of the service unit must be reduced from year to year in the
amount of the full accruing depreciation by a credit to depreci-
ation reserve; under the sinking-fund method, the public paying
return on the undiminished cost, there should be no deduction
from book value until the unit is retired or the sinking fund
matures, which times are theoretically the same. Thus, the
sinking-fund method becomes, so far as the property account is con-
cerned, the same as the replacement method of maintaining property.
Unfortunately, this has not been understood clearly, and the sinking-
fund method of accounting has not proceeded under this theory; con-
fusion has resulted, and, indeed, is likely to result, from any attempt
to account for depreciation under this method.

The Entries Under the Compound-Interest Method are, and are
explained, as follows : Income in the amount of the depreciation
allowance is charged to operating expense to keep the income state-
ment correct. Depreciation reserve is credited in the same amount
to keep the property statement correct. We now have assets consisting
of property depreciated by the depreciation reserve and cash or other
assets equivalent to that allowance. The depreciated property will earn



1474 VALUATION OF PUBLIC UTILITIES

fair return the next year, and the allowance also should earn fair return.
It may be — and should be, if possible — at once re-invested in the prop-
erty, but, however it is used, unless retained as working capital, cash



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