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ESTIMATING REPLACEMENT COST OF FIXED ASSETS
James C. McKeown and Robert E. Verrecchia
#453
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
December 2, 1977
ESTIMATING REPLACEMENT COST OF FIXED ASSETS
James C. McKeown and Robert E. Verrecchia
#453
Estimating Replacement Cost of Fixed Assets
James C. McKeown and Robert E. Verrecchia*
*Professor and Assistant Professor, respectively, Department of
Accountancy, University of Illinois at Urbana-Champaign.
Abstract
Various authors have argued for the use of the replacement cost valuation
basis In accounting reports with such success that official bodies have
either recommended or required that replacement cost figures be reported.
This success is primarily due to the effectiveness of the conceptual argu-
ments advanced for the use of replacement cost. Unfortunately the methods
advocated for estimating the replacement cost of fixed assets are not as well
developed as the conceptual arguments. First, we will review these methods in light
of the authors* stated or implied desire to measure replacement cost as either the
current cost of the asset in its current condition, or the availability of services
equivalent to those currently contained in the asset. Then we will propose a
method which will allow estimation of this replacement cost even in those situations
where there is either no used asset market or the used asset market which exists
is not sufficiently organized to allow ready estimation of used asset costs.
Finally, we will then compare our method for estimating replacement cost to
those previously proposed. We hope to point up that there are two salient
advantages to our approach: it subsumes the work of previous proposals, and
it permits greater generality.
Various authors have argued for the use of the replacement cost valuation
basis in accounting reports with such success that official bodies have
2
either recommended or required that replacement cost figures be reported.
This success is primarily due to the effectiveness of the conceptual argu-
ments advanced for the use of replacement cost. Unfortunately the methods
advocated for estimating the replacement cost of fixed assets are not as well
developed as the conceptual arguments. First, we will review these methods in
light of the authors' stated or implied desire to measure replacement cost as
either the current cost of the asset in its current condition, or the avail-
ability of services equivalent to those currently contained in the asset.
Then, we will propose a method which will allow estimation of this replacement
cost even in those situations where there is either no used asset market or
the used asset market which exists is not sufficiently organized to allow
ready estimation of used asset costs. Finally, we will then compare our
method for estimating replacement cost to those previously proposed. We
hope to point up that there are two salient advantages to our approach:
it subsumes the work of previous proposals, and it permits greater generality.
In their first detailed discussion of their choice of replacement cost,
Edwards and Bell (1972) conclude that "current cost" — cost currently of
Primary examples are: Bedford (1965), Edwards and Bell (1961), and
Revsine (1973).
2
Two examples are: American Institute of Certified Public Accountants,
Study Group on the Objectives of Financial Statements, (1973), pp. 41-43,
and Securities and Exchange Commission, (1976).
-2-
acquiring the inputs which the firm uses to produce the asset being valued —
is the appropriate replacement cost concept (pp. 91-92). However their
discussion at this point concentrates on valuation of inventory. In their
subsequent discussion of the valuation of fixed assets (which are not
usually "produced" by the firm), they make it clear that they believe the
RC should be based on the current cost of acquiring the existing asset in
its current condition (p. 175, p. 186n) . However they also suggest that
this valuation may be impractical since they recommend that current cost
of fixed assets be estimated at replacement cost new less accumulated
depreciation (p. 186). (Nonetheless, this recommendation is qualified
since they base it on the assumption that an accurate depreciation
method is being used.)
In his original discussion of replacement cost, Revsine (1973) states:
"Replacement cost balance sheet values represent the amount that a firm
would have to pay, as of the balance sheet date, in order to replace the
assets shown in the statement or to satisfy reported liabilities" (p. 69).
Although this is fairly general and he does discuss the problem of choosing
a depreciation method, his later statements and examples imply that he
regards the acquisition cost new less depreciation as a surrogate for the
current acquisition cost of the asset in current condition. (E.g., p. 100,
"Realizable cost savings are equal to the change in the market price of
assets held during the period." This will only be true if the "market
price" is the acquisition cost of fixed assets in current condition).
Although Bedford (1965) does not specifically deal with the distinction
between replacement cost new less depreciation and replacement cost of services
-3-
equivalent to those contained in the asset in its current condition, he co-
authored a later statement implying that his concept of replacement cost
would be "cash or cash equivalent that would have to be paid now to acquire
resources capable of providing services equivalent to those currently expected
to be extracted from the asset." (Bedford and McKeown, 1972).
From this examination it seems that the consensus of the literature is
that the objective of an accounting system based upon replacement cost
would best be met by relating those fixed assets currently held as directly
as possible to the market. This would mean that those methods which esti-
mate replacement cost as the cost new less depreciation should be viewed as
surrogates for the former measure. This consensus view is supported by the
definition of replacement cost given by the Objectives Study Group of the
AICPA: "A valuation basis quantifying assets (and usually liabilities) in
terms of present prices for items equivalent in capacity and services."
(AICPA, 1973, p. 41).
A major advantage of the approach to measurement of replacement cost
of fixed assets to be proposed in this paper is that it does not require
selecting a depreciation method. Only in this way can the replacement cost
system avoid arbitrary allocations. Any attempt to measure replacement cost
of fixed assets as the replacement cost new less depreciation would be an
arbitrary allocation and therefore subject to the same criticism that is
applied to historical cost systems (Thomas, 1969, pp. 89, 91).
Previous Approaches
The approaches which have been proposed for estimation of replacement
costs of fixed assets are of two basic types:
-4-
1. Estimation of replacement cost new then depreciating to a book
value. The replacement cost new can be estimated by direct
reference to the new asset market [Revsine (1973), p. 77, Edwards
and Bell (1961), pp. 185-7], price indexing [Revsine (1973), p. 77,
3
Edwards and Bell (1961), pp. 185-7, Brinkman (1977), p. 46-4]
or expert appraisal (including engineering estimate) of current cost
new [Revsine (1973), p. 77, Edwards and Bell (1961), pp. 185-7,
Brinkman (1977), p. 46-4].
Under these suggested methods, the depreciation is usually
computed from this estimate using a conventional accounting
depreciation method. Since most conventional accounting depre-
ciation methods are straight-line or accelerated and most theoretical
calculations are accelerating (higher depreciation in later years),
use of a conventional depreciation method applied to an estimate of
replacement cost new seems unlikely to yield a good estimate of the
market value (known or unknown) of the used asset. Therefore we
must reject this method using conventional accounting depreciation
methods.
Edwards and Bell (1961, pp. 175-176) suggest a depreciation
method based on study of the patterns of decline of second-hand
asset market values. This method should yield quite accurate results
if specific second-hand asset value are available. However in a
"thin" (relatively inactive) or non-existent used asset market,
3
Brinkman especially notes the necessity of assuming the adjustment
of index used to allow for technological change. [Brinkman, pp. 46-28
through 46-34] .
-5-
the accountant would be unable to get good estimates of the pattern
for specific assets. So although this method will probably perform
well in the presence of a well organized used asset market, it
provides little, if any, help in the absence of that market.
Both Edwards and Bell (1961, pp. 176-177) and Weil (1977,
pp. 46-35 to 46-43) discuss methods which are quite similar to the
annuity or sinking fund depreciation methods. These are based on
use of the internal rate of return and assume equal return from the
asset (or its replacement) during each year of its life. These
depreciation methods are probably the best of those proposed and if
used consistently would yield better approximations to the replacement
cost of the asset in the used asset market (or to the theoretical
estimates of what that value would be if there were a market) than
use of conventional depreciation methods. In fact the distinction
Weil calls this method functional pricing. He does not apply it
consistently to all five cases in his example. The reader may note that
Weil could have applied the same method used in cases IV-V to Cases
I and II yielding replacement cost of functional capacity in current
condition of $12,339 and $7,386 respectively.
The solutions to Cases I and II which would be consistent with
Weil's solutions to the other cases would be:
Cost new
■J present value of annuity
10 periods at 10%
+ operating cost of new asset
x ratio of capacities
- operating cost of
existing asset
x present value of annuity
5 periods at 10%
replacement cost of
existing capacity
I
II
$20,000
$20,000
*6. 14457
6.14457
$ 3.254.91
$ 3,254.91
+ 1,100
1,100
x 700
700
700
x 1000
$ 4,354.91
$ 3,048.43
- 1,100
- 1,100
$ 3,254.91
$ 1,948.43
x 3.79079
x 3.79079
$12,338.67
$ 7,386.11
-6-
between this method and the one that will be proposed in this
paper is that the former fails to consider the additional flexibility
inherent in owning (or keeping) an asset with fewer remaining
years of life.
2. Appraisal of the existing asset to estimate directly the current
cost of replacing the asset in its existing condition. Depending
on the accuracy of the appraisal, this approach could yield very
good estimates of the replacement cost of the asset in its current
condition. However, since the appraisals are likely to be most
accurate in those cases where the used asset market is active and
organized, the pattern method of depreciation suggested by Edwards
and Bell should also work well and with less expense. On the
other hand, in cases where the used asset market is not well-
organized, the appraisals would probably tend to be less accurate
and more costly. In general this approach would appear to be
practical only in the case of very large assets if it is
practical at all.
Since the approaches above either do not approximate the current cost of
the asset in its current condition, work only in the presence of a well-
organized used asset market, or ignore the increased flexibility of owning
assets with shorter remaining lives, it seems appropriate to propose an
approach which does not suffer from these deficiencies.
Definition of Replacement Cost
In order to provide a rigorous definition for the analysis and
development of estimation methods, consider the relationship between the
purchase price of new and used assets when an orderly used asset market
-7-
exiats. We will regard the purchase price of an asset as including the
full cost necessary to put the asset into service. In an orderly used
asset market we would expect the purchase price of a used asset to
"perfectly adjust" relative to that of a new asset such that the expected
cost associated with purchasing the used asset is identical to the ex-
pected cost associated with purchasing a new asset. For example, assume
that the purchase price of a new asset is $10,000, the purchase price
of a used asset is X, and a firm wishes to acquire services that can be
performed by either asset. If X were too high (low), the firm would
determine that the expected cost associated with paying $10,000 for the
new asset was lower (higher) than the expected cost associated with paying
X for the used asset. Thus the market would induce X to decrease (increase)
until the expected costs were identical. This perfect adjustment of the
purchase price of a used asset relative to that of a new asset will be the
basis of our definition. Specifically, we will define the replacement cost
of a used asset to be that price at which the expected costs associated
with "purchasing" the used asset which the firm currently holds, or pur-
chasing a new asset (at a known price), are identical.
Proposed Approach
In practice, an orderly used asset market which perfectly adjusts
prices may not exist. Nonetheless, we will use the definition of replace-
ment cost suggested above to derive these prices. Two different situations
will be considered: (1) one in which some sort of used asset market exists
in that used assets can be purchased or sold, but the prices at which these
transactions can be arranged are not easily observable; and (2) one in which
no used asset market exists at all — used assets cannot be purchased or sold.
-8-
The need to distinguish between these two market situations is
based upon two factors. First and most obvious, if a used asset market
does exist, the asset can be sold at some future date if a decision is made
to discontinue its use. This means that the "cost" of abandoning an asset
with a relatively large proportion of its life remaining is lower than if
there were no used asset market. Thus the added flexibility associated
with holding used assets is reduced. (The amount of this difference is
related to the amount which can be recovered from sale of a discarded asset.)
The second distinction between the two types of market situations is
that if there is no used asset market, the firm cannot buy a used asset.
Thus at the end of the life of an old asset, the firm can only choose between
either abandoning the use of this type of asset or buying a new asset. This
will be explored in more detail when the no used asset market situation is
discussed below.
The Used Asset Market Model
Assume the firm holds a used asset with k remaining years of life. This
asset could be replaced by a new asset with N years of life at cost P„ (k<N) .
Each asset performs the same level of services at the same cost during each
year of life and each is worthless at the end of its life. Either asset can
be sold at any time for a fraction, 5, of its replacement cost at that time.
The firm assesses the probability that it will abandon the use of this asset's
services at the end of any year (given the use of the asset during the year)
as 6. Thus we have:
P„ - the purchase price of a new asset with N years of life remaining.
(P„ is assumed known.)
P, ■ the purchase price of a used asset with k years of life remaining,
1 <_ k < N. (P, is assumed unknown. )
_9-
B » the fraction of the replacement cost for which a used asset may
be sold, 0 <_ B <_ 1.
BP, ■ the price for which a used asset with k remaining years of
life could be sold.
i = the appropriate discount rate for the firm
6 ■ the probability that the firm will abandon use of this asset's
services at the end of any year given that the services were used
during that year, 0 <_ 8 < 1.
On the basis of our definition, the replacement cost of the asset with
k years of life remaining can be derived (see Appendix) to be:
k fBP,
P - I {[
'k-£
1+i
e + p ] — - -^ --J u-e)*"1} (l)
This equation can be explained easily. The term in brackets is the
cost avoided at the beginning of year £ by having an asset with k years
of life remaining on hand. That is, if the company uses this asset it will
avoid paying the price of an asset with one year of life remaining, P..
In addition it will realize the exit value of the asset, BP,., if use of
the asset is terminated at the end of year I because this is what the
asset can be sold for. Assuming that use of the asset is not terminated
before year A, the probability the company will terminate it at the end
of year £ is 0, and the factor to discount to the beginning of year I
is ~rr±' The remaining factors compute the probability the use of the
asset is not terminated before year & and discount costs incurred at
the beginning of year % back to the balance sheet date.
The equation (1) has several intuitively appealing properties:
1. As the ratio of exit value to replacement cost, B, decreases,
the replacement cost, P, , increases relative to the price of
the new asset, 2 . That is, the significance of the added
-10-
flexibilicy which results from buying the used asset (with fewer
years of life remaining) increases as the proportion of replacement
cost realized from the sale of an unneeded asset decreases. An
alternative way to view this is that the penalty for having to
dispose of an asset before it is fully utilized becomes larger
as B becomes smaller.
2. As the probability, ©, of discontinuing use of the asset in
any year decreases, the replacement cost of a used asset, P, ,
decreases relative to the cost of a new asset, P . This is
consistent because the flexibility of holding the used asset becomes
less important as the probability of discontinuing use of the asset
decreases. In fact, when 8=0, the ratio — is equal
N
to the ratio of the present value of the annuities due (for the
given discount rate) for k and N years respectively. That is:
?fc x. A(k,i)
PN A(N,i)
where: A(m,i) ■ present value of annuity due for m periods
discounted at the rate of i per period.
This is the result Edwards and Bell (1961, pp. 176-7)
get and Weil (1977) should get for his Case I. Thus if the value
of flexibility is 0 (i.e., the firm will never discontinue use
of the asset), the adjustment for flexibility is 0 and the result
is identical to those suggested approaches which ignored flexibility.
3. If the probability, 6, of discounting use of the asset is 0
and the discount rate, i, also 0, the ratio of prices is:
PN~N
-II-
This is also the result which would be obtained if straight line
depreciation were used. (Accelerated depreciation would have an
even lower PjV^n ratio.) While a probability of termination,
8, of 0 might be reasonable in some cases, it is unlikely that a
discount rate of 0 is appropriate for any case. Thus, we must
conclude that use of straight line (or accelerated) depreciation
applied to the replacement cost new, P , will understate the
replacement cost of the used asset.
To illustrate the application of this approach, consider the following
situation (Case I from Weil (1977), p. 46-30):
Cost new - $20,000 «= P„
N
Life new ■ 10 years = K
Remaining life ■ 5 years ■ k
Discount rate = .10 = i
Probability of termination in any year ■ .10 = 6
Exit value/replacement cost = . 75 = B
Weil did not have this parameter specified. His solution is equivalent
to assigning value of 0 to 9. The sensitivity of the results of this value
of 6 will be examined later.
Weil did not need this parameter. Examination of equation (1) will show
that if 6 ■ 0, the value of B is irrelevant. A value of B of .75 is true
of some of the better organized markets, but typical values of B would
probably be considerably lower — particularly since we are assuming here
that we are not dealing with a well-organized used asset market. The
effect of the value of B will also be examined later (even to the extent
of considering the case where no used asset market exists).
-12-
The replacement cost of this asset is $12,927. That is, if a market
existed in which this used asset could be obtained, the management of this
firm would be indifferent between paying $20,000 for the new asset and
paying $12,927 for the used asset. Weil's solution (with 6=0) should
have been $12,339.
Having dealt with the case where the replacement asset is the same as
the used asset (no technological change), the obvious question is: What
happens if there has been technological improvement and a new asset is
available in improved form? The answer is derived by considering the forms
which the improvement could take. The primary possibilities appear to be:
longer operating life, increased capacity, or lower operating costs.
1. Longer operating life does not require any change in the
previously stated approach. The previous derivation did not
assume the original life of the new asset was equal to the life
of the new asset. The replacement cost of an asset is not
affected by the number of years of previous use, only by the
o
number of years of remaining use. Therefore, if the new
asset has a longer useful life than the original life of the
used asset, the life of the new asset is simply N and the
remaining life of the used asset is k.
No claim is made that the determination of P, is a simple 30 second
computation with a hand calculator. The details of solution are not
shown here simply to avoid boring the reader. The contention is made,
however, that the solution is straightforward and can be (and was)
determined by a simple computer program—or even one of the more
powerful hand calculators.
o
Of course for a given asset, the longer the past use, the shorter the
remaining use. The point is that the longer past use affects the replace-
ment cost only if it is tied to remaining use.
-13-
2. Increased capacity of the new asset Is considered in some detail
by Weil (1977, pp. 46-36 to 46-37). His discussion there applies
here as well. There are two subcases: the indivisible case
where the firm can not make use of the increased capacity, and
the divisible case where the firm can make use of the increased
capacity (either through use of fewer machines, rental of service
to external entities, increasing the operating life, etc.). We
feel strongly that the divisible case should be assumed. (It
seems likely that a company which held the asset with larger
capacities would receive benefit from the increased capacity.
Furthermore, the indivisible case would require reporting on the
balance sheet a replacement cost representing a larger capacity
than that currently available to the company. This seems inap-
propriate.) Under the divisible case the simplest way to adjust
Equation (1) for the difference in capacity is to simply multiply
the price of the new asset by the ratio of the used asset capacity
to new capacity before entering the replacement cost new into the
solution:
Vu
P ■ P • —
N N V„
N
where P ■ adjusted replacement cost new (to be used in solution)
PN = full replacement cost of new asset with larger capacity
V = capacity of used asset
V„ = capacity of new asset
Under the divisible case, the capacities of assets whose
remaining lives are between those of the used and new assets
-14-
do not affect the replacement cost of the used asset. The only
information needed is the capacity of the currently held asset
and the capacity of the new asset.
3. Operating cost decreases require a more complex adjustment than
the two preceding types of technological improvements. (It should
be mentioned that it is unlikely that a capacity change would be
made without a change in operating cost.) The complicating factor
is that the operating cost saving is only effective in those years
in which the asset's services will be used by the firm. Since we
are assuming that there is some probability the firm may discontinue
use of the asset, there is similarly some probability that the cost
savings of some future years will not be realized. Thus, equation
9
(1) must be modified to:
k E6Pk-il 1 k 1 - 8 £-1
A=l
9
Equation (2) requires the assumption that the total cost of acquiring
and operating an asset with one year of life remaining is constant over
the life of the new asset, That is as the operating cost decreases,
the cost of acquiring the asset increases. This assumption should be
valid so long as additional unexpected technological change does not
occur. Equations (1) and (2) can be simplified for computational purposes
(although some intuitive interpretability may be lost) to:
and:
-15-
where c. » operating cost of an asset which had i years of
life remaining at the valuation date, but the cost of
which is measured when the asset has j years of life
remaining. This cost is assumed discounted to the
' i
beginning of the year.
All other variables are as defined for equation (1).
Please note that equation (2) handles not only the case where operating
cost for the new asset is different from (presumably lower than) the operating
cost of the used asset, but it also allows for situations where the operating
cost for either asset is different for different years of that asset's life.
Therefore through use of equation (2) we are able to drop the assumption —
made for equation (1) — that the services of the assets are provided at the
same cost for each year of their lives. The solution is general as far
as pattern of operating cost is concerned.
Armed with these adjustments, it is now possible to compare the results
obtained under the approach proposed here with the approach proposed by
Weil. Table 1 presents the calculated results for the independent cases
considered by Weil (1977, p. 46-36) first under Weil's method (assuming
probability of discontinuing use of asset is 0), then under a variety of
combinations of e (probability of discontinuing use of the asset in any
year) and B (ratio of exit value to replacement cost). The values used for
9 are .10, .03, and .02 which correspond to expected number of years of
use of the asset of 10, 33 1/3, and 50 years respectively. (Expected
number of years of use is -r. ) A value of .10 is probably fairly high
for a stable industry, but might be appropriate for an industry where product
lives and processes change rapidly. Values of .75, .40, and .00 for B cover
-16-
the range of reasonable values. The value of ,75 is probably too high
since the markets we are considering are not well-organized. The middle
value (.40) is also somewhat high for a poorly organised market. Alterna-
tively, the value .00 represents the situation in which a firm would realize
nothing on the sale of a used asset. This might occur for one of two reasons.
Either there is no market for used assets or the expense of selling a used
asset is likely to be greater than the amount that will be recovered in a
sale. An illustration of the latter case would be when firms find it very
expensive to use a broker to find likely purchasers. This situation would
probably occur when a whole industry was changing product lines or processes
since the number of prospective sellers of used assets would be far greater
than the number of prospective purchasers.
Examination of Table 1 discloses the relationships mentioned above:
replacement cost of the used asset increases with increasing 6 and decreasing
£. Please remeraber that the cases represent independent situations where
different replacement assets are available- The various columns are presented
so that the reader may see the effect of the methods as applied to situations
where the replacement asset differs from the existing asset in different ways
(operating cost, capacity; years of life). Consideration of the different
columns demonstrates that there is a difference between an assumption of
6 = .00 and a G even as small as .02 (50 expected years of use). This
provides strong support for the use of the proposed method rather than one
which requires the assumption that 8 * .00.
No Used Asset Market
Situations where no used asset market exists can be handled by using
equation (1) or (2) above with E set equal to .00. However it is possible
-17-
Table 1
Existing asset:
operating cost:
Results of Weil's Cases
5 years of life remaining, capacity: 700 units,
$1,100
Replacement Assets
I II III IV
Cost New
Life when new = N
Capacity
Operating cost
Cost new of
capacity
existing
'N
Operating cost for
existing capacity
Weil's method*
§20,000 $20,000
10 years 10 years
700 units 1,000 units
$ 1,100 $ 1,100
$20,000 $14,000
$ 1,100
770
$12,339 $ 7,386
$20,000 $20,000 $20,000
10 years 12 years 12 years
700 units 700 units 1,000 units
$ 1,000 $ 1,100 $ 1,000
$20,000 $20,000 $14,000
$ 1,000 $ 1,000
700
$11,960 $11,127 $ 6,273
P (6 -
Pc(6 =
Pf(6 -
P^(6 =
Pc(6 -
P^(e -
pf(e -
Pc<e -
p|(e -
.1.
B =
.75)
$12,927
$ 7,854
$12,565
$11,843
$ 6,481
.1,
B -
.40)
13,739
8,495
13,399
12,832
7,622
•it
B -
.00)
14,634
9,199
14,318
13,920
8,477
.03,
B -
.75)
12,516
7,527
12,142
11,342
6,444
.03,
B -
.40)
12,763
7,723
13,396
11,643
6,682
.03,
B ■
.00)
13,045
7,946
12,685
11,986
6,954
.02,
B »
• 75)
12,457
7,480
12,081
11,270
6,387
.02,
B «
.40)
12,622
7,611
12,251
11,471
6,546
.02,
B -
.00)
12,810
7,761
12,445
11,700
6,728
This table is adopted from Weil (1977), p. 46-36.
*These are the results from Weil's method reported in his paper as modified
in footnote 4. This is also Pr(9 *.00).
This is also P5(9
@
The cost new of existing capacity and operating cost are each multiplied
by the following ratio: existing capacity/capacity of new asset. Also since
the operating cost given by Weil was assumed to occur at the end of the period,
the operating costs inputed into equation (2) were those shown discounted to
the beginning of the period (divided by 1.10).
-18-
to make use of the absence of a used asset market to develop an approach
which allows some generalization of the conditions regarding the probability
of discontinuing use of the asset. Lack of a used asset market means that
despite the firm's preferences it will both be unable to sell or buy a used
asset. Thus the replacement cost of the existing asset may be computed by
comparing the firm's only two alternaties: "buy" the existing asset or
"buy" a new asset. Furthermore, when either asset expires, it may only be
replaced by a new asset. To deal with this case, define A to be the prob-
ability that a new asset just purchased will not be replaced. This would
occur because the firm's need for the asset's services has ended by the end
of the new asset's life. That is, 1 - A is the probability another new
asset will be purchased when this one expires. Similarly, define A' to be
the probability that the existing asset currently in use will not be re-
placed. Then 1 - A' is the probability a new asset will be purchased when
the one currently in use expires. In this case, the replacement cost of
an asset with k years of life remaining is given by (proof is in Appendix) :
1 _
(1
1
- A')
x —
(1
+ i)k
1 -
(1
- A) )
(1
+ i)N
P. - : : pn (3)
Equation (3) yields exactly the same result as Equation (1) with B set
to .00 in those situations where equation (1) applies — that is where
the probability of abandoning the service in a given year is constant
over time (proof in Appendix). However Equation (3) can be used in many
situations where the probability of abandonment in a year is not constant
over time. All that is required is that A, the probability of abandoning
-19-
the asset's services within the lifetime of the (new) asset just purchased,
remain constant. The distribution of probability between years is not
constrained. In particular it may be that the probability of discontinuing
use of an asset's services will depend on the age of the asset. As an
asset grows older, the services it provides are more likely to be abandoned
because the services have a higher probability of reaching obsolescence
and because a smaller portion of the cost of the asset (with remaining life)
would be lost. This latter point is particularly important since in
the no used asset market situation the amount recovered from an abandoned
asset is zero.
In a similar fashion to Equation (1), Equation (3) can handle techno-
logical improvements such as increased life and capacity. However, as
stated, Equation (3) cannot handle operating costs which are not constant.
It could be modified to handle different operating costs, but this is
hardly seems worthwhile since it would be necessary to specify the year
by year distribution of probabilities within A and A".
Equivalent Services or Identical Asset
Two distinct concepts of the objective of replacement cost measurement
of fixed assets have been proposed. Edwards and Bell (1962, p. 196n)
clearly favor measurement of the cost of replacement of the identical asset.
Bedford (1965) and Revsine (1973) favor measurement of the cost of acquisition
of services equivalent to those contained in the existing asset. The approach
proposed above can be applied to either concept.
If the replacement cost of the identical asset is desired, the new
asset whose price, life and operating cost are identical to the existing
asset should be used as the standard. This may be difficult if such an
-20-
asset ie not available new, but this problem is to a certain extent inherent
in the identical asset concept. (A suggestion for handling this problem will
be made below. )
If the equivalent services concept is desired, the price, life and
operating cost of various new assets which could provide services equivalent
to those provided by the existing asset should be used as the standard.
(There may, of course, be problems in the identification of those assets
which provide equivalent services.) This concept of replacement
cost would appear to require that the lowest of the replacement cost
calculations (based on various new assets) be used as the measurement of
replacement cost. This would be consistent with the assumption that if
the firm were to replace the existing capacity with an asset capable
of providing equivalent services, its management should select the mode
of replacement which has the lowest cost.
An interesting feature of the proposed method of estimation is that
the estimate of the replacement cost of the existing asset would be the
same whether the identical asset or equivalent services concept is followed.
This equality would occur because firms which are considering both alternatives
(replacement with the same or an improved asset) would presumably buy whichever
one is "underpriced". This action across the market would thus adjust the
prices of the two (or more) new assets so that the expected costs associated
with each are identical. If this is the case, the estimate of replacement
cost will be the same when computed using any of the new assets which provide
equivalent services as a standard. (In fact the estimation approach proposed
could be used to compute perfectly adjusted prices for the various assets.)
Therefore the recommended procedure would be to use the parameters
of the identical asset new (if it is available new) as a basis for the
-21-
computation. This recommendation is made not because the identical asset
approach is favored on theoretical grounds, but simply because if two
methods yield the same result, the easier one should be used, and the easier
method here is obviously the one that avoids wherever possible the problem
of identifying assets which provide equivalent services. If the identical
asset is not available new, the parameters of an asset which can provide
equivalent services should be used in the computation. Again the result
should be the same as that which would be obtained if the parameters for
an identical new asset were known and used. Thus the proponents of the
identical asset concept can arrive at the estimate of the replacement
to
cost of the identical asset when no used asset market exists and the
identical asset is not available new.
Summary
An approach has been proposed which builds on the work of previous
authors to develop a method of estimating the replacement cost of an asset
in its current condition or the replacement cost of the services which can
be provided by that asset even in the absence of readily available used
asset market prices. The proposed approach allows adjustment for the
probability of discontinuing use of the asset's services as well as
technological changes such as increase in asset life, increase in asset
capacity or decrease in operating costs. Although used asset market prices
should be used if readily available, it is recommended that the proposed
approach be used in other cases for estimation of replacement cost under
either the identical asset or equivalent services concepts.
-
-22-
References
American Institute of Certified Public Accountants, Study Group on the
Objectives of Financial Statements, Objectives of Financial Statements,
"Report of the Study Group on the Objectives of Financial Statements,*'
Vol. 1, (New York: AICPA, 1973), pp. 41-43.
Bedford, Norton M. , Income Determination Theory; An Accounting Framework
(Reading, Mass.: Addison-Wesley, 1965).
Bedford, Norton M. and James C. McKeown, "Comparative Analysis of Net
Realizable Value and Replacement Costing," The Accounting Review, XLVII
(April, 1972), pp. 333-338.
Edwards, Edgar 0. and Philip W. Bell, The Theory and Measurement of
Business Income (Berkeley, California: University of California Press,
1961).
Revsine, Lawrence, Replacement Cost Accounting (Englewood Cliffs, N. J.:
Prentice-Hall, Inc., 1973).
Securities and Exchange Commission, Accounting Series Release No. 190
(SEC, 1976).
Thomas, Arthur L., "The Allocation Problem in Financial Accounting Theory,"
Studies in Accounting Research No. 3 (Evanston, 111.: American Accounting
Association, 1969).
Weil, Roman L., "Functional Pricing" in Sidney Davidson and Roman L. Weil,
Handbook of Modern Accounting, Second Edition (New York: McGraw-Hill Book
Company, 1977), pp. 46-35 to 46-43 .
;. . • ' ■■;
i
- ;
!
.;■•■■ *
APPENDIX
1. Prices of Current Assets When a Used Asset Market Exists.
Let
T+k
p0 ■ The purchase price of a used asset which becomes available
on the market T years from the balance sheet date, (at which
time it had k years of potentially useful life, l<k<N»
0<T) but which only has I years of life remaining, £<k.
T+k
(p0 is, in general, assumed to be unknown).
N
PN ■ The purchase price of a new asset which is available at the
balance sheet date and has N years of life remaining. (pN
is assumed known).
B ■ The fraction of the replacement cost for which a used asset
may be sold, 0<B<1.
T+k
Bp. ■ The price for which a used asset which became available T
years from the balance sheet date, was used for k-l years,
and has i years of life remaining can be sold.
■ The appropriate discount rate for the firm.
- (1+i)"1
The probability that the firm will abandon use of this asset's
services at the end of any year given that the services were
used during the year, 0 <_ 6 < 1.
-2-
T+k
c. e The operating cost of an asset which became available T years
from the balance sheet date and originally had k years of
potentially useful life, but the cost of which is measured
when the asset has i years of life remaining. This cost is
assumed discounted to the beginning of the year.
For convenience, superscripts were suppressed in the earlier discussion.
However, by defining P. , l<kfN, as
Pk = Pk '
the analysis in the body of the paper will be consistent with that in the
appendix. Finally, it will be assumed that for all T>0,
T+k T+k 1,1
Pl 1 - pl 1 *
This assumption simply states that the purchase price plus operating cost
of a used asset with one year of life remaining remains constant over time.
This is not an unreasonable assumption concerning a short lived used asset,
and will considerably facilitate the analysis.
Suppose that the firm assesses the probability that the asset's services
will be terminated T years from the balance sheet date, 1<T<«>, to be
T-l
6(1-6) . Then the cumulative probability is
T
Jt-1
Probability of termination in T or fewer years ■ E 6(1-6) .
Jt=l
Thus the probability of termination in T or fewer years is distributed as a
geometric distribution with unknown parameter 6 and moment generating function
-3-
M(t)
6e
[1- (1-6) e']
A stream of purchases of new and used assets is any combination of purchases
such that a firm can secure the service of one, and only one, of these assets
in any given year. The price system will be derived by assuming that the
expected cost associated with any conceivable stream of purchases over the life
of the asset's service is equal to the expected cost associated with all other
alternative streams. To derive these prices, consider two possible streams
of purchases. Both streams are identical until T years from the balance sheet
date, at which time a used asset with k years of useful life remaining is purchar
in the first stream, while in the second an asset with one-year-of-life remaining
is purchased annually from T years from the balance sheet date until T+k
years. After the end of the year T+k, both streams continue to be identical.
Since these two streams are identical except between years T and T+k, the
expected cost associated with both will be identical if and only if the expected
costs which result from the purchase decisions between years T and T+k are
identical. For example, in the first stream the expected cost that results
from the decision to purchase a used asset with k years of life remaining T
years from the balance sheet date is
T+k
E
£=T+1
T+k _T
Pi, Q
V P1+k + E cT+k (J*"1
H^ pT+k-£ * , cT+k+l-j **
6(1-8)
A-l
T+k J£ T+k T+k ftj-l
?k Q j=T+l T+k+1~^
(1-6)
T+k
(Al)
-4-
The first term in (Al) is the cost when the use of the asset's services is
terminated at the end of year I times the probability of this occurring,
summed over all I between T+l and T+k. The second expression is the cost
times the probability the use of the asset's services will not be terminated
before the end of year T+k. Similarly, the expected cost associated with
purchasing an asset with one-year-of-life remaining between years T and T+k is
T+k
E
£=T+1
J-T+l
{p3 Q^1 + c{ Q?'1 }
0(1-6)
£-1
T+k
E
j=T+l
{ P
J-l
+ c
J rJ-1
9(1-6)
T+k
(A2)
with the same logic applying.
Equating these two expected costs yields
T+k
E
£=T+1
f
T+k _T __*. T+k * T+k _j-l
Pk Q - BQ pT+k_£ + ^ ^T+k+l-j Q
8(1-9)
l-l
T+k
Prk qt +
j-T+l
cT+k Q3"1
T+k+l-j H
(1-6)
T+k
T+k
E
£=T+1
I { V\ Q3"1 + c\ Q3"1 }
J-T+l
9(1-6)
l-l
T+k
E { p3 Q3'1 + c\ Qj-1 }
j=T+l
(1-6)
T+k
-5-
Eliminating a factor of {Q(l-0)} from both sides and letting m=£.-T yields
T+k
k
2
m=i
_nm T+k
BQ p,
e(i-e)
m-1
k
+ Z
m=*l
m
J-l
{pWj qJ-1 + cT+j Qj-1
cT+k Q^"1}
Ck+l-j 4 '
6(1-6)
m-1
-1 T+k _j-l
- c
k+l-j
.4 QJ >
U-e)1
Reversing the summation signs permits
T+k „ _,,m T+k a/, 0»m-l
>k = E, BQ Pk-in e(1-9>
m=l
k
+ Z
j-l
{p^ + c^
k+l-j ; *
z e(i-e)in"1 + (i-e)k
m»j
But since
z eu-e)0"1 + (i-e)k -
m-j
e ed-e)1""1 + z eu-e)111""1 = i e(i-e)111"1
m=j
m=k+l
m=j
(l-e)3"1
T+k
k
E
m=l
{BQ6} P£k + pT*n
T+m T+k
+ c, - c
k+l-m
{Qd-e)}^1
Finally the assumption that for all T>0,
-6-
T+l T+l
Pi + ci
1 1
Pi + ci
implies
T+k
k
E
m=l
<BQ8> p™ + p\ + c\
- c
T+k
k+l-m
{QCl-e)}0"1 (A3)
As a means of simplifying (A3), let us propose that
T+k
E {B6Q + Q(l-e)}
m*!
k-m
1.1 T+k
p, + c- - c
rl 1 m
(A4)
We will demonstrate that (A3) implies (A4) using an inductive argument. When
T=0, k=l, (A3) implies
1.1 1
Pi + cl - cl
This is consistent with (A4). When T=l, k«l, (A3) implies
1 1
Pi + cl
- c.
This is also consistent with (A4). When T=0, k=2, (A3) implies
?2 ' {B6Q} t?1 + {p* + cj -c*} + {p* + cj - c*} ,
but since p. *
1 1
Pi + c,
- c,
{B6Q + Q(l-9)}
P1 + c
- c.
1 j 1
Pt + ci
- c.
-7-
£ {B8Q + Q(l-8)}
tn=l
2-m
1,1 2
p- + c. - c
rl 1 m
Therefore (A3) Implies (A4) when k«l, 2, T+k=l,2. Thus suppose that (A4)
holds for p. . :
p£ = Z {BQ6 +Q(l-8)}(k-1)-m fpi+cj-
m=l *■
m
We will show that this along with (A3) implies
Jk
I {BQ6 + Q(l-e)}
m-1
k-m
1.1 I
pl + Cl " Cm
From (A3)
{BQ9} p^ + Pi + cj-c^^l {Qd-9)}m-
ef3! v
Z
m=2
{BQ9}p^m + Pi + cj-c*,.^
{Q(i-e)}
m-1
+ {BQ8} p^_1 + p\ + c\ -c*
Z {BQ8} p^_m + pj + c\
m=2
- c
k+l-m
{Q(l-8)}:
m-2
+ {BQ8} p^ + p* + c* - c
-8-
k-1
(Q(i-e)} i [ {BQ0} p*^^ + PJ + oj - cl(k_l)+1^
(Qd-e))3"1
+ {BQ6} pfc-1 + Pl + e1 - ck
But using (A3) again ,
{Q(l-e)} p*_x + ' {BQ6} p£_x + p*+cj
- c.
- {Q(l-e) + BQ6} p^_1 + p* + c* - c£
k-1
- {BQ0 + Q(l-e)} E {BQ6 + Q(l-e)}
m=l
(k-l)-m
1,1 £
b 1 1 B J
+ {P1 + Cl - Ck}
k-1
E {BQ6 + Q(l-fl)}
m=l
k-m
1,1 £
p, + c. - c
*l 1 m
,1,1 £,
+ CPl + c± - ck>
k
e {BQe + Q(i-e)}
art
k-m
1,1 £
p- + c. - c
rl 1 m
£ o
Thus if (A4) holds for p. - , (A3) Implies that (AA) holds for p. .
We will simplify our expression one more time. (A4) implies that
W N XT
p™ = E {BQ9 + Q(l-6)}N"m
W"l
1.1 N
p, + c. - c
rl 1 m
-9-
Thus p.. , which is unknown, can be written in terms of all known parameters;
N
Pw "
1
pj- I ({BQ9} + Qd-e))**
1 N
c. - c
1 m
N
E ({BQ8} + Q(l-6))
m=l
N-m
Therefore p, can be written in terms of all known parameters by inserting
the above expression for p. in (A4) and rearranging terms:
m=l
1 N
ci ~ c™
l m
}{ E Zk^}
m=l
N
I
m=l
N-m
+ E Z1^ c£
m=l *•
- c
m
where Z - ({BQ6} + Q(l-S)).
However, if Z < 1,
k
E
npsl
k-m
k-1
z*
m=0
1-Z
1-Z
Therefore
'k
rl-Z ■, , N
N
E
m=l
N-m
1 N 1
} + { E Z
m=l
k-m f 1
m
(A5)
Similarly, if Z - 1 (i.e., 6=0, i=0)
E Zk-ffi = k.
m=l
Thus
, MN/-1 ,. > k
*'«»S- I,ci-<£ } + ! «
m=l ' m=l
ci " cm
J. m
(AS')
-10-
2. Prices of Current Asset When a Used Asset Market Does Not Exist.
Let
P, ■ The purchase price of a used asset which is currently
available and has k years of useful life remaining,
l<k<N. (P, is assumed unknown)
P„ » The purchase price of a new asset which is currently
available and has N years of useful life remaining (P„ is
assumed known and does not change over time).
i = The appropriate discount rate for the firm.
Q = d+i)"1
A" * The probability of abandoning the asset's services within
k years, conditional on the fact that a used asset with
k years of life is purchased, 0 <_ A" < 1.
A ■ The probability of abandoning the asset's services within
N years, conditional on the fact that a new asset with N
years of useful life is purchased, 0 <_ A < 1.
Costs will not be introduced, nor will superscripts, since it will be assumed
that the price of a new asset remains fixed over time. Formally, p„ = P.,
for all T>0.
There are only two possible purchase streams when a used asset market
does not exist: initially "purchase" a used asset with k years of life
remaining and buy new assets thereafter, or purchase a new asset initially
-11-
and buy new assets thereafter. If a used asset with k years of life remaining
is purchased initially, the firm assesses A" as the probability that the
T-l
asset's services will be terminated within k years, and A(l-A"*)(l-A) as
the probability that the asset's services will be terminated between years
k+N(T-l) and k+NT, 1<T. However, if the firm purchases a new asset initially
(and thereafter), it assesses the probability that the asset's services will
T-l
be terminated between years N(T-l) and NT, 1<T, as A(l-A)
Equating the total expected cost of these two purchase streams enables
us to derive prices:
oo £
Pv + E E PM Qk+jN A (1-A') (1-A)*
K £=0 j-0 N
00 £
E E P QJN i(l-A)£
S,=0 j=0
Rearranging terms yields
oo £
P, - [1-Qk(l-A')l E E P QjN A(l-A)*
£*0 j=0
oo £
E E
1=0 j=0 j=0 i-j
Finally, recall that E E <^N A(l-A)£ - E QjN E A(l-A)£
E <^N (1-A)j - [1 - QN (1-A)]"1. Thus
J-o
m [1 - q* (1-A')] (A6)
K [1 - QN (1-A)] N
-12-
Although (A6) was derived separately, we can show that It Is equivalent to
(A5) under appropriate assumptions:
1) B=0, which implies no used asset market
2) c. * c, all costs are constant
T+N N
3) P« = PN - PN for all T>0, the price of a new asset remains
fixed over time.
k
E
J-l
N
4) a' = e e (l-e)^"1
5) a - e e (l-e)^"1
J-l
Under these assumptions (A3) implies
p = pk= {1 - (Q(l-6))k} p
k k {l - (Q(i-e))N} N
_ (I - Qk d-e)k} p
U-QN(i-e)N} N
But since by assumption
k k
l-A' - l- e ed-e)^""1 = i-e e (l-e)^"1
J-l J»l
= !_ e [l- (i-e)k] m k
1 l- (i-e) u e; »
-13-
and similarly
1-a - (i-e)N ,
it follows that
P „ [i-rv-n*] p
k [1-QN(1-A)N] N '
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