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Optimal tax depreciation
Journal
of Accounting
and Economics
OPTIMAL
2 (1980) 213-237.
Recerved
Publishing
Company
TAX DEPRECIATION
Lee MACDONALD Tl~r Uniousit~
North-Holland
oj Rochesrer.
WAKEMAN* Rochester. .V I’ 14h27. L’S.4
May 1979, final vcrston
recetved December
1980
Currently, solutions to the choice of depreciatton methods for tax purposes arc obtained through numerical simulation. This paper presents a simple capttal budgeting technique whtch. tn conjunction with Descartes’ rule of stgns, analyttcally derives the opttmal regular depreciation method under the existing U.S. tax code. This techmque is then used to determtne the optimal depreciable life for short-lived assets and, finally, the analysis IS extended to cover the choice of depreciation methods under the Class Life Asset Deprectation Range System.
1. Introduction and summary The Internal Revenue Code of 1954, in liberalizing the rules concerning depreciation, allowed either the double declining balance method or the sum of years-digits method to be used as alternatives to the straight line method for tax purposes.’ In the first article to discuss this‘choice, Davidson and Drake (1961) compared the net present value of depreciation-associated tax savings for the two accelerated methods and derived the optimal depreciation method for various combinations of asset life and discount rate. Since then, as Stickney (1980) points out, most of the literature in the area has followed Davidson and Drake, using both the same decision rule and the same technique of simulation. ’ None of these subsequent papers presents a *I wish to thank. wtthout Implicating, Mark Walsh for wrtting the computer programmes whtch underly the results of section 3, Philip Meyers and Clyde Stickney for gutdrnp me through the labyrinth of Internal Revenue Procedures, and Jean-Marie Gagnon. Mtchael Jensen, Rtchard Leftwich, John Long. Richard Ruback, Tom Russell. Brian Rutherford, Bill Schwert. Clifford Smith and Marty Weingartner for theu trenchant comments. I am also Indebted to the referees and the edttors of thus journal for the many tmprovements that they suggested. Funding for this paper was provided by the Center for Research tn Government and Busmess, The Unrverstty of Rochester. I am also grateful for the support provided by the European Instttute for Advanced Studies m Management, Brussels. Thts research originated partially tn work undertaken whtle the author was a fellow at the Internattonal lnstttute of Management. Berhn. ‘See appendix 1 for detatls of these methods. ‘For example. Davidson and Drake (1964) Schoomer (1966). and Roemmich. Duke and Gates (1978) discuss the choice between the double declimng balance and sum of yearsdigtts methods for regular deprectation; Greene (1963), Ricks (1964) and Messere and Zuckcrman (1979) determine the optrmal time to swatch from declinmg balance to straight hne depreciation; Schwab and Ntcol (1969) and Sunley (1971) analyse the switch from the double declining balance
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Publishing
Company
214
L. Macdonald Wakeman, Optimal tax depreciation
rigorous analysis, and as a result they perform unnecessary computations3 and occasionally advance solutions which are in error.* An easy way to analyse the relative present values of different depreciation methods is to consider them as mutually exclusive projects, with equal investments, whose positive cash flows are the reductions in corporate taxes paid. Since these projects have equal lives, an incremental analysis can be used, in conjunction with Descartes’ rule of signs,5 to determine the optimal depreciation method under the existing U.S. tax code. The purpose of this paper is to present this technique. Besides reducing the number of computations involved and thus the chance of making errors, this procedure is likely to be easily adaptable to future changes in the U.S. tax code or to non-U.S. tax laws. In addition, the approach illustrates the application of standard capital budgeting techniques to tax planning problems. Section 2 sets out the regular depreciation methods allowed under the present procedures of the United States Internal Revenue Code. Incremental analysis is used in section 3 to determine the optimal depreciation method and to estimate the costs of an incorrect choice. for regular depreciation,6 Section 4 first applies this technique to the trade-off between the investment tax credit and depreciation cash flows for short-lived (less than 7 years) assets, and then extends the analysis to the Class Life Asset Depreciation Range System. Section 5 provides a synopsis of the main conclusions.
2. Allowable regular depreciation methods Until 1954, the Internal Revenue Service allowed only the straight line method for tax purposes. In this method, an annual depreciation rate equal to the reciprocal of the allowed asset ‘life’ is applied to the depreciation base (equal to the ‘adjusted cost’ minus the allowed ‘salvage value’ of the assets),’
method to the sum of years-digits method allowed by the Internal Revenue Procedure of 1967; Stickney and Wallace (1975), Crumbley and Hasselback (1976). and Hasselback and Orbach (1979) hetermine the choice of depreciation methods under the Class Life Asset Depreciation Range System; and Ford (1973), Byars (1979). and Osteryoung. McCarty and Fortin (1979) discuss the choice of optimal asset life given the investment tax credit. ‘Byars (1979) simulates the results ‘for 25,920 different sets of input parameters’. ‘For example, the year to switch from double declining balance to sum of years-digits given in Messere and Zuckerman (1979) is consistently one year too late. Schwab and Nicol (1969) make several mistakes in their table 1, and Osteryoung, McCarty and Fortin (1979) err in their estimates of optimal depreciable lives. 5For an extended discussion of this topic see. for example, Burnside and Panton (1960). ‘In sections 2 and 3, the analysis concentrates on new personal property depreciated under the regular (non Class Life Asset Depreciation Range) system. ‘Since 195X the adjusted cost is equal to the accounting cost less additional first year depreciation (AFYD) of 20”/;, of the cost of personal property with a life of at least six years (although the IRS limits the additional first year depreciation to $2000). Since 1962, a reduction in salvage value of up to IO::, of the cost of assets with lives greater than or equal to 3 years has been allowed. These points are analysed in section 4.
Wakeman, Optimal tar depreciation
L Macdonald
215
providing an even pattern of annual tax savings over the life of the asset. Thereafter, the double declining balance’ and the sum of years-digits methods could be used for new personal property. The resulting choice of time patterns for the tax savings from depreciation is il!ustrated in fig. 1.
2c
.
-T-
7 Straight
ZC(l-s) T+1
; \’
. ~
\ ‘> t\ ‘\
Line
(dsL)
Sum of
Years-Digits
Double
Declining
(dtSYD)
-_-
(dtDDB)
Balance
_____
\ ‘\
’ \
\ \
\
\
\
\
\
\
\ x
l
\
L-a
\
--_--_
Annual Tax Savings
\ \ \ \
I Depreciation
Note: Notation:
in
Year
The area under all three curves savings from normal depreciation, C
:
Adjusted
s
:
Ratio
T
:
Allowed
T
: Marginal
*
:
Switch
is
equal to the total C(l-S)T
Cost of the Depreciable
of Allowed Useful
Salvage
Life
c
T
t
tax
Asset
Value to Adjusted
Cost
of the Asset
Tax Rate to Straight
Line Depreciation
Fig. 1. Annual tax savings from the three baw
depreciation
methods
‘After 1962, taxpayers were allowed to switch from the double declining balance method to the straight line method during the life of the asset without the permission of the Internal Revenue Service. This advantageous switch is incorporated in the analysis.
216
L. Mucdonuld
Wakernun, Optimul
tax depreciation
The set of regular depreciation methods was further widened in 1967 and 1974 to allow a taxpayer to start with the double declining balance method and switch to the sum of years--digits method or to start with the sum of years digits method and then switch to either the double declining balance or straight line methods.” The assumptions binderwhich the choice between these methods is analysed in section 3 are:
(i) The asset is either new personal
property, new residential rental section 1250 property, or section 1245 property. and is depreciated as a separate unit.“’ (ii) The sal\,age value, S. and the useful life, T. of the asset allowed in the Internal Revenue Code are given, and the asset is held for this life. (Earlier sale will not change the conclusions reached in section 3. but does affect the choice of optimal life discussed in section 4.) (iii) The appropriate tax rate is considered constant over the life of the asset. (A tax decrease will not affect the conclusions. If ;I reasonably large tax increase is anticipated the optimal strategy may be to start with the straight line method and switch to sum of years digits when the tax increase is enacted.)’ ’ (iv) The income generated each year is sufficient to cover the depreciation charged (the conclusions will still hold under the weaker condition that the ‘carry back’ and ‘carry forward’ provisions of the Code ensure that no depreciation-generated tax savings are lost, although the gains quoted herein will be smaller.)‘”
3. The optimal regular depreciation method Rather than exhaustively compare the preaent values of the tax savings generated by the depreciation methods, this section uses Descartes‘ rule of signs and the constraint that the depreciation totals for all methods arc “See KWXW Prvccdt~rc74-i 1. section IV, paragraphs (b) and (c), respecttvely. The Class Life Asset Ikprcc~ation Range System. introduced in 1971, uses essentially the stme depreaation methods and is discussed in section 4. “‘For a rewcw of the assets covered by these delinittons. set HorvitL (1Y7) ). The Income forcca~t method of dcprectation allowed in section 167(b). and generally assoctated with televtsion and firm operations, IS not discussed. Set Levin and Adess (lY74) for dctatls Nor does and deprectatton for utiltties. this paper analyze the interrelationshtp between regulation discussed tn Linhart (lY70). “See Gagnon (1976) for a detailed analysis of thts point. This analysis is also only strictly applicable to corporations and those individuals unalcted by the tax preference provistons of the Tax Reform Act of lY76. Schwartz and Livingston (lY78) and Brogdon and Fisher (iY78) show that, for certatn individuals, the tax preference penalttcs incurred outweigh the tax delay benefits of the accelerated depreciation methods. ‘lIf current losses are so large that the ‘carry back/forward’ provtsions would bc exhausted. the minimum depreciatton possible should be taken.
L. Macdonuld
217
Wakeman, Optimal tax depreciation
equal, over equal lives, to determine which method is superior. Table 1 summarises this analysis, which first proves that the accelerated methods are then shows that these accelerated superior to straight line depreciation, methods are subsets of the switching methods allowed by the 1974 procedure, and finally proves the superiority of the double declining balance switching to sum of years-digits method.
3.1. The choice
het~ren
uccclerated
Theorem 1. The uccelerutrd un_y p0sitit.e dixount rute.
methods
methods
dominate”
the strtright
Line method jkr
Proof: Consider first the sum of years-digits method. From fig, 1 we can see that the annual difference in depreciation, ~(d;“‘~-dS~), is first positive and then negative.ls By Descartes’ rule of signs, since there is only one change of sign there is at most one real discount rate, r*, which equates the present value of tax savings of the two methods.” But since the total depreciation for the two methods is equal, we know that r=O’),, is a valid solution. Hence zero is the only solution. For any positive discount rate (and for any salvage value), the present value of the positive incremental cash flows in the first half of the asset’s life will exceed that of the negative cash flows in the second half, and therefore the accelerated method dominates.16
Now consider the double declining balance method. The general pattern of incremental tax savings for (dffiB -dTL) is essentially similar to that for (dfYD-dfL), and the proof is identical. Theorem
2.
The
yeurs-digits c&e
choice
methods
depends
for
between
the
an usset
with
only upon the uppropriate
‘“In this paper, ‘A dominates B’ means net present value of tax savings generated positive discount rates. 14From
double
appendix
declining
a given discount
balunce
depreciable
und sum
rate.
that, under the assumptions set out in section 2, the by method A exceeds those from method B for all
1, we can show that
(dfYD-d~~)~C(1-s)~(T+1-2r)/T(T+1). “For
the internal
lbFurther.
equation:
CT=,
T-((d~‘D-d~L)/(l
it is simple to show that the difference ppo
is positive
rate of return
-pVSL=5.(C(1-ss)iT(T+1)).
for any positive term structure
of
/lye und salcuge
of discount
in present
i
,=I rates
+r)‘)-0.
value of tax savings,
(T+l-2r)/
I n’ (l+r,) r=,
,
L. Macdonald
218
Wakeman, Optimal tax depreciation
Proof The general pattern of the incremental tax savings is as shown in fig. 2.” Since there are two sign reversals in the incremental tax savings pattern, there are two roots that satisfy l/(l+r)~O. Since zero is one of these
Incremental Annual Tax Savings
/
1
/
// I / I /
+ T(dt
DDB _ ,s,,
\
/ I
\
)
Year
t
T
/
\ / \
/ \ \ \
Fig. 2. Incremental
“From
--
1’
tax savings between the double declining balance and sum of years-digits depreciation methods (for the case of zero salvage value).
appcndlx
1, we can show that
t=1,2 . .._.k,
(dFD” -
where k is the year for which the following condition
first holds:
L.Macdonald
Wukemann, Optimal Table
Relationships
between
the
present
Sum of Years-Digits switching to Double Declining Balance
i
$YD t
value
219
tax depreciation
I
of tax savings methods.
of alternative
regular
Declining Balance switching to SUTIof Years-Digits
Double
<
dDDB + SYD t
+ DDB
PI
v
Sum of Years-Digits
Sum of Years-Digits switching to Straight Line
I
Double
z
$.YD t
V
depreciation
Declining
Balance
dDDB t
function
of:
asset salvage discount
life value rate V
>
Straight
Line
dsL t
solutions, there can only be one positive discount rate P at which, for a given salvage and depreciable life, the present values of the tax savings from the double declining balance and sum of years-digits methodsI are equal.
The ioci of r* are presented in fig. 3 for various ratios of the salvage value to the cost of the asset, s. For a given salvage ratio, the sum of years-digits “When there present vahres is asset life, salvage values should be
are two or more sign reversals in the pattern of cash flows, the difference in not independent of the term structure of discount rates. For combinations of value and term structure which lie close to the loci shown in fig. 3, the present compared.
220
L. Macdonald Wakeman, Optimal tax depreciation
method dominates at lower positive discount rates, and at a higher discount rates the double declining balance method dominates. Given the asymmetric treatment of salvage value in the two methods, increasing the salvage value expands the range over which the double declining balance dominates (the
,s
_---
-
= 0
/-
DDB /
SYD
/ / .6
Discount Rate
I .5 I .4
.1
I
DDB
a---
’ II’
I
-
__-__-_-----
DDB ----_ SYD
1
10
5 = 0.05
20
s = 0.10
I
I
30
40 Asset
Life
* T
NOTES s:
Ratio of adjusted
the salvage cost of the
DDB:
The double declining in this region.
SYD:
The sum of years-digits this region.
value asset
to
balance
method
the
method
dominates
dominates
in
Fig. 3. The discount rate that equates the present value of tax savings of the double drclinmg balance and sum of years digits methods as a function of asset hfe and ratio of salvage value to adjusted asset cost.
loci in fig. 3 move to the southeast as the salvage ratio increases) until at relatively high salvage value/adjusted cost ratios (s~O.15), there is only one sign reversal in the pattern of incremental annual tax savings and the double declining balance method dominates for all discount rates.” 19For earlier discussions of this optimal choice problem, see Davidson and Drake (1961, 1964) and Schoomer (1966). For many countries [see ‘Principle 68’ in Price Waterhouse (1975)] these are the most accelerated methods available and therefore this is the appropriate decision.
L. Macdonald
3.2.
The choice
between
Wakeman, Optimal
switching
tax
221
depreciation
methods
Consider first starting with the sum of years-digits method and switching to either the straight line or double declining balance method. Theorem 3. Never struight line method.
switch
j’rorn
the sunl qf‘ yeclrs-digits
method
then
to the
Proof
Switching from the sum of years-digits method to the straight line method at the end of any year k (where k < T- 1) entails forgoing a tax saving now in order to gain an equal tax saving in the future.20 Thus for positive discount rates it will never be advantageous to switch. the sum of Fears--digits method is initiuted. sbtitch to rhc balance method ut the end of‘thut yeur k rjlhich is the lrqcr retllroot of the polynomial,
Theorem
double
4.
[f
declining
k2(l-s)-k(l+T)(l-s)+T(t+sT)=O. Proqf: The depreciation given by”’
while the depreciation given by
Setting
in year k+ 1, after switching
under
these depreciation
zoFrom
appendices
1 nnd
dSYD*SI._dSID=
I+ l.I
l+’
the sum of years-digits
flows to be equal
produces
at the end of year k. is
method
equation
in year k+ I is
(I), with two
2, we can show that
C(l-.\) -~-~~T(Tfl)
.(k+l-T)
for all
k
“Although the swatch to the double declining balance method allows the salvage be incorporated into the depreciable base. section 5.04 of the 1074 lnternal Revenue that:
value, S, to Code states
‘For changes in method of depreciatton to the declimng balance method, the rate of depreciation applicable to the property shall be based on the useful life of such property from the date of acquisttion, and not the expected remaining life from the date the change becomes effective’, and therefore
the depreciation
rate is 2/T rather
than 2/(T - k).
L. Macdonald
222
Wakeman,
Optimal
tax depreciation
roots, k,,,,, and kupFr. If we switch when k,,,,, $ k< kupperr the pattern of incremental tax savings (dSYD*DDB- dsYD) for t = k+ 1 to T will be first negative and then positive and therefore it will not be advantageous to switch. But if we switch when k = k,,,,Fr, the incremental cash flows will be maximising the present value of the first positive and then negative, remaining tax savings.22 Table 2 presents the values of k~,___~which maximise the present value of tax savings using the sum of years-digits++double declining balance depreciation method for various combinations of depreciable life and salvage value to adjusted cost ratio.23 Table
2
The year at the end of which a switch should be made from the sum of yearsdigits the double declining balance method.” Salvage value to adjusted cost ratio (s)
Depreciable 7
x
9
10
15
20
25
30
40
0
N
N N N D D
N N N 6 D
N N N 7 D
N
N N 18 15 D
N N 22 19 D
N N 27 23 D
N 3x 35 31 D
0.05 0.1 0.15 0.2
N N D D
life (T)
method
to
______
N
13 11 D
“N: Do not switch; the SYD method dominates the SYD+DDB D: Complex roots to the polynomial in k; the SYD-DDB DDB method (see fig. 3).
Now consider starting with the double switching to the sum of years-digits method:
declining
method. method IS dominated
balance
method
by the
and
Theorem 5. If the double declining balance method is initiuted, switch from the double declining balance method to the sum years--digits method at the end of the first year k for which
of
Proof: Although there is no gain in switching in the very early years (see fig. l), once the condition that tax savings are equal in the year following a “There is no point considering k > k,,ppcI since the present v&e will be reduced by Similarly, the optimal switching time when 05 k < k,,,,, is k=O. But this is equivalent to with the double declining balance method rather than the sum of years-digits method. 23This method is the optimal choice for assets acquired in the first half of the year if depreciated under the modified half year convention of the Asset Depreciation Range See section 4 for details.
waiting. starting they are System.
L. Macdonald
switch is met, then the declining balance-tsum methods for the remainder
Wakeman, Optimal
tax depreciation
223
differencez4 in tax savings between the double of years-digits and double declining balance of the project’s life., 2(1’1- 1 -t) ____~____ ~~~~~ (T-k)(T+l-k)
-c
t>li,
can generally25 be shown to be first positive and then negative, as illustrated in fig. 4. Therefore, the present value of the remaining tax savings will be
+
-
Fig. 4. Annual difference in tax savings between the double declinmg balance-sum digits and double declining balance methods, when the switch from double declining sum of yearsdigits is made at the end of year h.
of years balance to
‘JAlthough the switch to the sum of yearsdigtts. which takes place before a switch from the double declining balance method to the stratght line method. requires that the salvage value. \C. be excluded from the depreciable base, section 5.03 of the 1974 Internal Revenue Code states that ‘the rate of deprectation shall be based on the remammg useful life of such property’. and therefore the depreciation rate is 2/‘(T+ 1 -k) rather than 2(T - b),:? (T + I ). See appendtx 2 for detai!s. For an alternative view of the appropriate remaining useful hfe, see the discussion of the appropriate Belgian law m Gagnon and Broquet (1975). 25For non-zero salvage values, the pattern of cash flows can be first posittve. then negattve, and finally positive. See the note to table 3 for details.
224
L. Macdonald
Wakeman, Optimal
tnx depreciation
maximised if the switch from the double declining balance method to the sum of yearsdigits method takes place at the end of the first year for which @f:;S’D
Rearranging
> dDDB zzz k+,’
this equation
gives us. the condition
stated in Theorem
5.
Table 3 presents the values of the switch year, k, which maximises present value of tax savings using the double declining balance-+sum years-digits depreciation method for various combinations of depreciable and salvage value to adjusted cost ratio.26 Table The year
3
at the end of which a switch should be made method to the sum of yearsdigits
Salvage value to adjusted cost ratio(s) 0
0.05 0.1 0.15 0.2
Depreciable
life
the of life
from the double method.”
declining
balance
(T)
7
8
9
10
15
20
25
lb 2 3 N N
1
1 2 3 N N
1 2 3 N N
1 2 4 N N
1
2 3 N N
3 5 8/N’ N
1 3 5 9 N
30 1 3 6 10 N
40
1 4 7 13 N
“N: Do not switch; the DDB method dominates the DDB+SYD method. %ince the depreciation in the second year is equal under the DDB-SYD and DDB methods for zero salvage value, changing after the second year will have the same effect. ‘For this case, (d!‘Ds-sy“-dfYD) after the switch year is first positive, then negative and then again positive in years 19 and 20, i.e., there is a real root other than zero. For after-tax discount rates greater than or equal to 5.95 76, the DDB method dominates.
Having shown that all other depreciation methods are either inferior to, or subsets of the double declining balance-+sum of yearsdigits and sum of yearsdigits-+double declining balance methods, we have now only to compare these methods. Theorem 6. The double declining balance+sum of years-digits method dominates the sum of years-digits-tdouble declining balance method for all salvage ratios, all depreciable lives and all positive discount rates.
Proof: Consider yearsdigits-+double
first the case of zero salvage value. In this case the sum of declining balance method reverts to the simple sum of
Z6For assets under the half year convention, the values for k in table 3 should be Interpreted as the ji// year at the end of which the switch should take place. See appendix 4 for details.
Wakeman, Optimal tax depreciation
L. Macdonald
years-digits
method
(see table 2) and the annual 2(T+
df;D-DDB=dSYD=C.
I
1 -t)
?r(T+l)
depreciation t=l,2
’
225
is given by
,..., T.
The switch from the double declining balance method to the sum of yearsdigits method takes place at the end of year 1 (see table 3) and the annual depreciation is therefore given by dtq’B-=‘D = c
f
t=l,
t = 2,3,. . ., T
The incremental 2
(dtTB’syD - dfyD) is positive
depreciation
for the first year,
2
T’E7 and negative
thereafter
since
.2(T+l-t)<2(7-+l-t) (T-
1)T
T(T+l)
’
t = 2,3,. . ., T.
We can then conclude, using Descartes’ rule and the constraint on the total depreciation taken, that the double declining balance+sum of years-digits method dominates the sum of years-digits method for all positive discount rates. Consider now the case of positive salvage value. As the salvage value increases, the tax savings from the double declining method are unaffected in the earlier years.” However, since the depreciation base for the sum of years4igits method is reduced by the salvage value, the tax savings from switching to the sum of years-digits method are reduced, and the switch is increasingly delayed until eventually no switch takes place (see table 3). Similarly, the tax savings using the sum of yearsdigits-tdouble declining balance method are reduced in the earlier years and the switch to the double declining balance method is advanced (see table 2). The pattern of the incremental depreciation (df)DB’syD- dFrDADDR) is entirely similar to that in the zero salvage value case, although the absolute magnitude of these incremental flows is naturally larger and the initial period of positive tax savings is longer. We can therefore draw the same conclusion. “The method.
salvage
value is not deducted
from the adjusted
cost for the double
declining
balance
L. Macdonald Wakeman, Optimal tax depriation
226
To summarize
the analysis
to this point:
The optimal regular depreciation method for tax purposes is to start with the double declining balance method and to switch to the sum of years-digits method at the end of the year given in table 3.
3.3. The costs of incorrect
depreciation
choices
There are costs involved in implementing the double declining balance -*sum of years-digits method.‘* However, the consequences of an incorrect choice are by no means negligible: the use of the double declining balance method for a project costing two million dollars with a thirty year depreciable life, zero salvage value and a 5 % after tax discount rate would result in a present value of tax savings of $592,02929 compared to the $629,885 resulting if the double declining balance-tsum of years-digits method had been used - an entirely unnecessary loss of over 130/, of the initial investment. As can be seen in fig. 5a, when the straight line method is chosen the losses involved are rather large, increasing with increases in the discount rate, salvage ratio and depreciable life. The sum of yearsdigits method, which before 1974 was optimal for an asset with zero salvage value and a long depreciable life, is still relatively efficient for this case, but as can be seen in fig. Sb, the losses increase substantially with increases in the salvage ratio. In contrast, the losses incurred when the double declining balance method is used are quite large for long-lived assets with zero salvage value (see fig. 5c), but decrease with increases in the salvage ratio - to the point where, for salvage ratios exceeding 15 %, the simple double declining balance method is optimaL3’ The conclusion of this section is that the administrative costs of implementing the double declining balance+sum of years-digits methods would have to be very high indeed to outweigh the tax benefits of this method. 28Rev. Proc. 74-77 states: ‘Change in method of depreciation. Taxpayers may expeditiously obtain, under the prescribed conditions, the Commissioner’s consent to change certain methods of depreciation accounting to certain other methods by filing application on Form 3115, within the first 180 days of the taxable year the change is to become effective, with the Director of the Internal Revenue Service Center where the return will be tiled.’ Clyde Stickney notes that not only must notice be given to the IRS, but detailed information about acquisition cost, cost additions, depreciation taken in prior years, salvage value, and other items must be supplied for each asset. “Assuming
a 50% tax rate.
30Appendix 3 presents calculations of the present value of tax savings methods for various value of asset life, discount rate and salvage ratio.
of the alternative
L. Macdonald Wakeman, Optimal tax depreciation
r
50%
(a)
Straight
Line
221
Depreciation
Percentage Loss
Asset
(b) Percentage Loss
Life
Sum of Years-Digits
S = .lO
1
25%
--
___---
5
10
15
20
25
Asset 50%
_-
__--
*.________---
O%-
HA
/
r (c)
Double
30
35
s
= .os
=
0
S
=
0
s
se
.05
s =
.lO
40
Life
Declining
Balance
Percentage Loss
___
5
Note:
s = ratio
10
of
15
salvage
20
--_
----
25
30
Asset
Life
value
to
35
adjusted
40
cost
Fig. 5. Reduction in the present value of tax savings incurred by an incorrect choice of depreciation method, as a percentage of the present value of the double declining balance-tsum of years-digits method; tax rate = 50 %, discount rate = 10%.
228
L. Macdonald Wakeman, Optimal tax depreciation
4. Extensions The above analysis can also be applied to the choice of depreciation life and to analyse the Class Life Asset Depreciation (ADR) System.
4.1. The investment
optimal Range
tax credit
Under the investment tax credit programme, business firms can deduct, as that year. a credit against income tax, a percentage31 of this new investment For the full loo/:, credit to be taken, an asset must have a depreciable life of at least seven years. For assets with lives of live to six years, the credit is reduced to 0 of lo%, and assets with three to four year lives receive a credit of only 3 of loo/,. The question ‘What is the optimal econorkc life to choose?’ is beyond the scope of this paper since other cash flows, such as maintenance, will no longer be constant when comparing unequal-lived assets. However, we can determine which depreciable life3’ should be considered for each economic life. If the IRS allows depreciable lives shorter than the given economic life to be chosen, the appropriate cash flows per investment dollar are those given in table 4. Since the total cash flows per investment dollar are equal for the 3 and 4 year depreciable lives (i.e., 0.51333) and for the 5 and 6 year depreciable lives (i.e., 0.54667) and the sign of the incremental cash flows (e.g., 3 years minus 4 years) changes only once we can immediately conclude that the shorter lives dominate for all positive discount rates. ‘For the comparisons in which the total cash flows differ the incremental cash flows still change signs only once,33 and therefore we know that there can be only one positive discount rate which equates the net present values for the different depreciable lives. As shown in table 5, the longer life is optimal for all lower discount rates, and for any higher discount rate, the shorter life is preferred.3” The analysis for depreciable lives longer than the given economic life differs slightly since in this case the depreciation for the years beyond the asset’s economic life is taken at the end of the asset’s last year and the investment tax credit ‘excess’ will be recaptured when the machine is written “initially, in 1962, the percentage was 7 “i,. Under the current (1975) legislation, the credit is lo%, with an extra l-1$% being allowed for firms which meet certain criteria with respect to their employee stock option plans. 3ZTo the extent that the IRS permits. “Except the comparison of the 6 and 7 year depreciation lives for a 7 year economic life. The incremental cash flows in this case change sign twice, but we can show that rcr0 for one of the solutions. 34F6r example, an asset with an economic life of 5 years should be depreciated over a 5 year life if the discount rate is 11.4% or lower and over a 3 year life if the discount rate is higher.
L. Macdonald
Wakeman, Optimal Table
The after-tax”
cash flows per investment
Depreciable
dollar
229
tax depreciation
4
from depreciationb
and the investment
tax credit.
life 5
6
1
0.25867 (20/50r + Z/30)
0.22667 (151457 + 2/30)
0.23714 (421147~ + 3130)
0.12 (3/12~)
0.1152 (121507)
0.10667 (10/457)
0.09796 (30/1477)
008 (21127)
0.0864 (9!5or)
0.08533 (R/457)
0.08163 (25/147~)
0.04 (1112r)
0.0576 (6!5&)
0.064 (6/45r)
0.06531 (20/147r)
0.0288 (3/5Qr)
0.04267 (4/45r)
0.04898 (15/1477)
0.02133 (2/45~)
0.03265 (10/147T)
Year
3
4
I
0.35333’ (6/9r+ 11’30)
0.27333 (6/12r+
2
0.10667 (2197)
3
0.05333 (1197)
4
1,‘30)
5 6
0.01633 (5/147?)
I Total cash flow
0.51333 (7 + l/30)
0.51333 (T + l/30)
0.58 (T + 3130)
0.54667 (r.22130)
0.54667 (T + 2/30)
“A tax rate, T, of 48 y, is used bThe double declining balance *sum of yearsdigIts method is used. For this calculation, a zero salvage value is assumed. ‘For a 3 year depreciable life, the investment tax credit per investment dollar is f (lo%), and the first year depreciation under the double declining balance method is 2(1/3)r per investment dollar, for a total tax credit of 6/9t + l/30. For a tax rate of 48 %, this credit is worth 0.35333 dollars per investment dollar. Table The optimal
Economic life
depreciable
life for a given economic
Depreciable -
5 life as a function
of after-tax
discount
rate r.’
life 3
4
5
6
7 D
LL
r>O
D
D
D
SL LL
r>O -
D r>O
-
-
-
D
D
D
SL LL
r> 11.47” -
D -
rs 11.4’;b r>O
D
SL LL
r>11.4”,, _
D _
r<11.49/, -
D D
r>O
SL
r>12.7%
D
D
D
rs 12.7 %
_ D
“SL: If the IRS permits a depreciable life shorter than or equal to the planned economic life. LL: If the IRS only permits a depreciable life equal to or greater than the planned economic life. D: Dominated solution.
230
L. Macdonald
Wakeman, Optimal tax depreciation
OfF.3s Since there are no differences in the investment tax credit taken, the total cash flows are again equal for all comparisons. There is still only one incremental cash flow sign change36 and therefore the shorter life dominates3’ If for a given economic life any depreciable life is permissible, the solution is the same as that presented in table 5 for the shorter life option. Additional first year depreciation (AFYD) allows a taxpayer to shift up to 209; of the total depreciation (up to a maximum of $2,000 in a given year) from the last years to the first year3* for assets with lives of at least six years. Therefore AFYD favours the 6 and 7 year depreciable lives. However, since the additional depreciation is limited to the first $10,000 of investment, the impact of this provision on the choice of optimal life is limited to assets which cost less than $100,000.39 The conclusion is that for any planned economic life, asset cost, and discount rate, there is one optimal depreciable life which quite often differs from the economic life for short-lived assets.
4.2. The class life asset depreciation
range
(ADR)
system
The foregoing analysis can also be used to choose the optimal depreciation method and optimal depreciable life for assets which are depreciated under the ADR system. The ADR system, introduced in 1970, differs from regular depreciation in several important ways : (i) Salvage value is ignored for all depreciation methods. (ii) The depreciable life can be chosen in the range of the guideline life plus or minus 205;. (iii) The set of applicable depreciation methods is larger, since there is a
35For example, when comparing 5 and 7 year depreciable lives for a 5 year economic life, and remaining depreciation is taken in year 5 and, if a 7 year depreciable life has been chosen, f of I am indebted to Mark Wolfson for clarifying this the lo”,, investment tax credit is recaptured. pomt. “Again, the comparison between the 6 and 7 year depreciation lives for a 6 year economic life differs. There are three sign changes and therefore three roots. However, the two non-zero roots are both negative. Contrary to the results for all other comparisons. the 7 year depreciatton hfe dominates the 6 year life for all posittve discount rates. 37For example, for a 4 year economic hfe. the 4 year depreciable life dominates all longer depreciable lives for any positive discount rate. 381t IS normally optimal to apply the AFYD to the longest-lived asset purchased in a given year, AFYD affects the opttmal depreciable life choice only if less than $10.000 is spent on longlived assets in the year. 39For a $100,000 Investment, the normal first year depreciation for a 7 year life is given in table 5 as $23.714. Even with the additional first year depreciation of $2,ooO the total first year depreciation of $25,714 is less than the first year depreciation for a 5 year life of $25,867. Since the sign pattern is unchanged, the dominance results remain unchanged.
L. Macdoncrld
Wakeman, Optimal
231
tax depreciation
choice of first year conventions - the .half year convention the modified half year convention (MHYC).40
(HYC) and
Since Stickney and Wallace (1977), using computer simulation, survey the optimal choices of depreciable life, depreciation method and first year convention for the ADR system,4’ I will concentrate on those points where the analytical technique compliments or illuminates their simulation results. Consider first the half year convention (HYC) for ADR assets.42 The depreciation flows are equal to those taken under the normal half year convention, and since the appropriate salvage rate is zero, the optimal depreciation rule is simple: Start with double declining balance after the first full year of depreciation.
and switch
to sum of years-digits
The modified half year convention (MHYC) differs from the HYC convention in that assets acquired in the first half of the year get a full year’s depreciation in the first year while assets acquired later in the year receive no depreciation that year. The depreciation flows for assets acquired in the second half of the year under the MHYC are similar to those for the HYC, and lead to the same optimal depreciation rule: Start with double declining balance the beginning of the third year.
and switch to sum of yearsdigits
at
But the combination of the full first year depreciation and the extra half year of remaining life for assets acquired in the first half of the year under the MHYC favours the sum of years-digits43 method. The result is that the depreciation taken under the double declining balance+sum of years-digits method in the year after the optimal switch is actually less than the sum of years-digits depreciation for that year. Hence the analysis developed in Theorem 6 to prove the dominance of the double declining balance-tsum of years-digits method is no longer relevant. Stickney and Wallace (1975) go so far as to state that ‘the sum of yearsdigits method is preferable for all depreciable lives greater than four’. Analytically, this statement is incorrect since there are two changes in sign of the incremental depreciation, (@DB-sYD -,fYD), implying that there is at least one real discount rate, r*, other than zero, for which the net present values of the two methods are equal. The locus of these rates for differing asset lives has a shape which is similar to the s=O locus in fig. 3. However, r* is 16.1 “/o, ‘“For further details of the Class Life (ADR) System. see Feinschreiber (1975) or Stickney and Wallace (1977). “See especially their tables A-l to A-24 and B-l to B-22. “‘Assets receive a half year’s depreciation in the year of acquisition, and remaining life is set equal to (7--j ). See appendix 4 for details. “The depreciation m year r under the sum of years+digits method is now 2(T+ l.S- t)/ T(T+l)instead of2(T+l-t)/T(T+l)for t>l.
232
L. Mncdonald
Wakeman, Optimal
tax depreciation
41.9 ?,, and 59.6 y0 for the 4, 5 and 6 year asset lives respectively, and since Stickney and Wallace, quite reasonably, used a maximum discount rate of 20 “,, in their simulation, it is easy to see why they made the error, For practical purposes, their rule is optimal for assets acquired in the first half of the year under the MYFC with less than loo/;, salvage value.44 Since the analysis of the impact of the investment tax credit under the ADR system is similar to that described in section 4.1,45 I refer the interested reader to Stickney and Wallace for the results.46 The conclusion of this section is that analytical solutions for optimal depreciation strategies under the Class Life (ADR) System can be easily derived.
5. Conclusion
To date, solutions to the choice of optimal depreciation strategies for tax purposes have generally been found through numerical simulation. This paper presents a simple capital budgeting technique which is used to derive analytical optimal depreciation strategies4’ and which will probably be equally useful in analysing future changes in the Tax Code.
Appendix 1: Simple depreciation methods
The following C: S: s: T: A,:
notation
is common
to all methods:
Adjusted cost of the depreciable asset, Allowed salvage value of the asset, s/c7 Allowed useful life of the asset, Depreciable base of the asset at the end of year r, where t=O, 1,. . ., T.
44Although salvage value does not affect the depreciable base for the SYD method, it does change r* smcc it differentially affects the timing of cash flows in the last years. For example, if a salvage coefficient of .s=O.l is applied to the 5 year life, the depreciations under the double declining balance-sum of years-digits method terminates in year 4 rather than year 6 while the sum of years-digits method depreciation still terminates in year 4. Then the double declining balance-sum of years-digits method dominates the sum of years-digits method for all discount rates of 1‘I;, or more. The general effect of salvage value on the choice of optimal depreciation method for the MFYC is similar to the effect of salvage value on the choice between double declining balance and sum of yearsdigits illustrated in fig. 3. “Although the range of depreciable lives for a given guide-line life is limited to ~20’;,, around that guide line life. Mark Wolfson points out that there is a potential for even more flexibihty since asset classifications are not iron-clad. “For example, their Table A-2 shows that, under the HYC, an asset with a gmde-line of 4 years should be depreciate - over 5 years for discount rates of 12”;, or less and over 3 years for discount rates of 13 :‘., or higher. “‘With the bonus of explaining why a certain depreciation strategy is optimal under the given clrcurnstances.
L. Macdonald Wakeman, Optimal tax depreciation
A-l .I. Straight
233
line depreriution
The initial depreciable Salvage Value, AEL=C(l
base
equals
the Adjusted
less the Allowed
-s). taken in year t under the straight
The depreciation
Cost
line method
is therefore
and AFL by A;L=C(l
A-l 2.
Sum
-$$!,
qf yews-digits
t==O,l,...,
7‘.
depreciation
A;YD=C(l
-s),
dsy”=C(l t
3(T+ -.s).-rcr+F,
and
A;”
A-1.3.
Double
1 -t)
(T-thT+ = C ( I- s ) m--(T+r-.
declining
balance
t-l,.?,...,
1 -t)
T
t=O,l,...,T
depreciation
A switch from the double declining balance method to the straight line method is allowed, and takes place at the end of the first year, k, for which the tax saving in the following year from switching to straight line is greater than that from staying with double declining balance. Before the switch,
L. Macdonald Wakeman, Optimal tax depreciation
234
The switch is made condition first hoi&: dDDB’SL f,k
at the end
> dDDB =t
of the year
k for which
the following
9
i.e.,
After the switch,
Appendix 2: Switching between accelerated methods A-2.1. Double declining
balance to sum of years-digits
The depreciation taken under switch from the double declining given by
the sum of years-digits method after a balance method at the end of year k is
2(T-k+l-t) ~T_k~~T_k+l~~
-
t=LT...,(T-kL
since under section 5.03 the remaining life of (T-k) is the relevant depreciable life, and salvage value must be deducted to get the depreciable base for sum of years-digits. Thus,
t=k+l,
k+2 ,..., T.
235
L. Macdonald Wakeman, Optimal tax depreciation
(The problem of a switch from double declining balance to straight line before a switch from double declining balance to sum of years-digits can be disregarded since the tax savings in the year after the switch, k + 1, will be greater for a switch to sum of years-digits than for a switch to straight line).
A-2.2. Sum of years-digits to double declining balance The depreciation taken under the double switch from the sum of yearsdigits method d;;D-DDB = C (1 -s)
declining balance method after a at the end of year k is given by
(T-k)(7-+1-k)+s T(T+
1)
t=k+l,k+2
,..., T.
Appendix 3: The net present value of alternative depreciation methods Throughout this appendix, it is assumed that the initial value of the asset is $1,000,000. After-tax Present Values can be obtained by multiplying the figures presented by the appropriate tax rate. Table A.1 Present value of depreciation stream; salvage value/cost ratio = 0.0 (initial value of asset: rsl,OOO,OOo). Life of asset
Discount rate
SL
SYD
DDB-SL
DDB+SYD
5
0.05 0.10 0.15
865895 758158 67043 1
89403 1 806142 132315
896505 810998 739427
899866 816437 746094
10
0.05 0.10 0.15
772174 614457 501877
828460 700988 603785
816919 685282 587557
831214 705612 609692
20
0.05 0.10 0.15
623111 425678 312966
717885 546973 436211
68925i 515387 408780
719119 548879 438492
30
0.05 0.10 0.15
512415 314231 218866
629142 442432 335971
592029 409107 311278
629885 443505 337198
40
0.05 0.10 0.15
428977 244477 166044
557096 368548 271205
516634 338254 251379
557602 369241 271972
L. Macdonald Wakeman, Optimal tax depreciation
236
Table stream;
A.2
Present
value of depreciation
salvage value/cost rsl,OOO,c0O).
ratio=O.l
(initial
value of asset:
Life of asset
Discount rate
SL
SYD
DDB+SL.
DDB+SYD
5
0.05 0.10 0.15
779306 682342 603388
804628 725528 659138
817306 747564 688099
817306 747564 688099
10
0.05 0.10 0.15
694956 553011 451689
745614 630889 543407
751793 641912 558085
754694 645866 562209
20
0.05 0.10 0.15
560800 383111 281670
646096 492276 392590
644747 494575 398603
654166 504331 406567
30
0.05 0.10 0.15
461174 282808 196919
566228 298189 302374
561529 398994 301665
514279 40929 1 314441
40
0.05 0.10 0.15
386080 220029 149440
5Ot 386 331698 244084
495659 333274 250060
509527 342119 354934
Appendix 4: The regular half year convention The regular half year convention considers all acquisitions to be made at the middle of the taxable year. Thus, the depreciation taken in the first half year is 5 of the first year’s depreciation, that taken in the second year (first full year) is ) of the first year plus ) of the second year, etc. The condition for a switch from double declining balance to sum of yearsdigits at the end of the full year (k=O, 1,2,. . ., T) is
References Brogdon, William J. and Ted. L. Fisher, 1978, Accelerated depreciation v. the minimum tax, Taxes, Sept., 530-534. Burnside, W.S. and A.W. Panton, 1960, The theory of equations, Vol. I (Dover Publications, New York). Byars, Richard B., 1979, Selecting an optimal asset life to maximize tax related cash flows: A computer simulated study, Taxes, Feb., 101-109. Crumbley, D. Larry and James R. Hasselback. 1976, The asset depreciation range system, The CPA Journal, Jan., 29-33.
L. Macdonald
Wakeman,
Optimal
tax depreciution
231
Davidson, Sidney and David Drake, 1961, Capital budgetmg and the ‘best’ tax depreciation method, Journal of Business, Oct., 4422452. Davidson, Sidney and David Drake, 1964, The ‘best’ tax depreciation method - 1964, Journal of Business, July, 258-260. Feinschreiber, R., 1975, Tax depreciation under the class life ADR system (Amacom, New York). Ford, Allen, 1973, Selecting a useful life for investment credit. Management Accounting, April. 14-16. Gagnon, Jean-Marie. 1976, A note on tax depreciation, Journal of Business Administration 7. no. 2, Spring, 117-124. Gagnon, Jean-Marie and Claude Broquet, 1975, La politrque d’amortissement fiscal de la firme, Recherches Economiques de Louvain 41, no. 1, March, 3-23. Greene, ED., 1963, Changing from declining-balance to straight-line depreciation, The Accounting Review, April, 355- 362. Hasselback, James R. and Kenneth N. Orbach, 1979, Two dectsion rules for determining the optimal switching points under ADR, Taxes, Nov., 774.780. Horvitz, Selwyn A., 1971, Problems and benefits in workmg with deprectable property after the tax reform act of 1969, Tax Lawyer 24. no. 3, Sprmg, 5699588. Levin, Jack S. and Melvin S. Adess, 1974, The income forecast deprectation method: Its current usage: Problems and outlook. The Journal of Taxation, Feb., 70-75. Linhart, Peter, 1970, Some analytical results on tax depreciation, Bell Journal of Economics and Management Science 1, no. 1, Spring, 82-112. Messere, Carl J. and Cilroy J. Zuckerman, 1979, A generalised approach to the timtng of optimal depreciation switches, Taxes, June, 3899393. Osteryoung, Jerome S.. McCarty. Daniel E. and Karen Fortin, 1979. A note on the optimal tax lives for assets quahfymg for the investment tax credit. The .i\ccounting Review. April, 301306. Price Waterhouse, 1975, Survey of accounting princtples and reporting practices in 46 countrres (Price Waterhouse International). Ricks, RB.. 1964, Year to swatch to straight-line deprectation, The Accounting Review. July. 685-688. Roemmich, Roger, Duke, Gordon L. and William H. Gates, 1975, Maximistng the present value of tax savings from depreciation, Management Accounting, Sept., 55~ 57, 63. Schoemer, B.A., 1966, Optimal depreciation strategies for income tax purposes. Management Science 12, no. 12, Aug.. B452B479. Schwab, Bernhard and Robert E.G. Nicol, 1969, From double-declining-balance to sum-ofyears’-digits depreciation: An optimum switching rule, The Accounting Review, April, 292296. Schwartz, Sheldon and Richard J. Livingston, 1978. What prtce accelerated depreciation?, Journal of Real Estate Taxation, Fall, 3245. Stickney, Clyde, 1980, A note on optimal tax depreciation research, Unpublished worktng paper (Dartmouth College. Hanover, NH). Stickney, Clyde P. and Jeffrey B. Wallace, 1975, A guide to increasing the maximum available depreciation deductions under the ADR system, Taxatton for Accountants 14, no. 1, July, 42-48. Stickney, Clyde P. and Jeffrey B. Wallace, 1977, A practical guide to the class life (ADR) system (Lawyers and Judges, Tucson, AZ). Sunley, Emil M., 1971, An optimum switch from double-declining balance to sum-of-the-years-digits depreciation, The Accounting Review, July, 574582.
of Accounting
and Economics
OPTIMAL
2 (1980) 213-237.
Recerved
Publishing
Company
TAX DEPRECIATION
Lee MACDONALD Tl~r Uniousit~
North-Holland
oj Rochesrer.
WAKEMAN* Rochester. .V I’ 14h27. L’S.4
May 1979, final vcrston
recetved December
1980
Currently, solutions to the choice of depreciatton methods for tax purposes arc obtained through numerical simulation. This paper presents a simple capttal budgeting technique whtch. tn conjunction with Descartes’ rule of stgns, analyttcally derives the opttmal regular depreciation method under the existing U.S. tax code. This techmque is then used to determtne the optimal depreciable life for short-lived assets and, finally, the analysis IS extended to cover the choice of depreciation methods under the Class Life Asset Deprectation Range System.
1. Introduction and summary The Internal Revenue Code of 1954, in liberalizing the rules concerning depreciation, allowed either the double declining balance method or the sum of years-digits method to be used as alternatives to the straight line method for tax purposes.’ In the first article to discuss this‘choice, Davidson and Drake (1961) compared the net present value of depreciation-associated tax savings for the two accelerated methods and derived the optimal depreciation method for various combinations of asset life and discount rate. Since then, as Stickney (1980) points out, most of the literature in the area has followed Davidson and Drake, using both the same decision rule and the same technique of simulation. ’ None of these subsequent papers presents a *I wish to thank. wtthout Implicating, Mark Walsh for wrtting the computer programmes whtch underly the results of section 3, Philip Meyers and Clyde Stickney for gutdrnp me through the labyrinth of Internal Revenue Procedures, and Jean-Marie Gagnon. Mtchael Jensen, Rtchard Leftwich, John Long. Richard Ruback, Tom Russell. Brian Rutherford, Bill Schwert. Clifford Smith and Marty Weingartner for theu trenchant comments. I am also Indebted to the referees and the edttors of thus journal for the many tmprovements that they suggested. Funding for this paper was provided by the Center for Research tn Government and Busmess, The Unrverstty of Rochester. I am also grateful for the support provided by the European Instttute for Advanced Studies m Management, Brussels. Thts research originated partially tn work undertaken whtle the author was a fellow at the Internattonal lnstttute of Management. Berhn. ‘See appendix 1 for detatls of these methods. ‘For example. Davidson and Drake (1964) Schoomer (1966). and Roemmich. Duke and Gates (1978) discuss the choice between the double declimng balance and sum of yearsdigtts methods for regular deprectation; Greene (1963), Ricks (1964) and Messere and Zuckcrman (1979) determine the optrmal time to swatch from declinmg balance to straight hne depreciation; Schwab and Ntcol (1969) and Sunley (1971) analyse the switch from the double declining balance
0165-4101/8OjOOOO-0000!$02.50
cc:, North-Holland
Publishing
Company
214
L. Macdonald Wakeman, Optimal tax depreciation
rigorous analysis, and as a result they perform unnecessary computations3 and occasionally advance solutions which are in error.* An easy way to analyse the relative present values of different depreciation methods is to consider them as mutually exclusive projects, with equal investments, whose positive cash flows are the reductions in corporate taxes paid. Since these projects have equal lives, an incremental analysis can be used, in conjunction with Descartes’ rule of signs,5 to determine the optimal depreciation method under the existing U.S. tax code. The purpose of this paper is to present this technique. Besides reducing the number of computations involved and thus the chance of making errors, this procedure is likely to be easily adaptable to future changes in the U.S. tax code or to non-U.S. tax laws. In addition, the approach illustrates the application of standard capital budgeting techniques to tax planning problems. Section 2 sets out the regular depreciation methods allowed under the present procedures of the United States Internal Revenue Code. Incremental analysis is used in section 3 to determine the optimal depreciation method and to estimate the costs of an incorrect choice. for regular depreciation,6 Section 4 first applies this technique to the trade-off between the investment tax credit and depreciation cash flows for short-lived (less than 7 years) assets, and then extends the analysis to the Class Life Asset Depreciation Range System. Section 5 provides a synopsis of the main conclusions.
2. Allowable regular depreciation methods Until 1954, the Internal Revenue Service allowed only the straight line method for tax purposes. In this method, an annual depreciation rate equal to the reciprocal of the allowed asset ‘life’ is applied to the depreciation base (equal to the ‘adjusted cost’ minus the allowed ‘salvage value’ of the assets),’
method to the sum of years-digits method allowed by the Internal Revenue Procedure of 1967; Stickney and Wallace (1975), Crumbley and Hasselback (1976). and Hasselback and Orbach (1979) hetermine the choice of depreciation methods under the Class Life Asset Depreciation Range System; and Ford (1973), Byars (1979). and Osteryoung. McCarty and Fortin (1979) discuss the choice of optimal asset life given the investment tax credit. ‘Byars (1979) simulates the results ‘for 25,920 different sets of input parameters’. ‘For example, the year to switch from double declining balance to sum of years-digits given in Messere and Zuckerman (1979) is consistently one year too late. Schwab and Nicol (1969) make several mistakes in their table 1, and Osteryoung, McCarty and Fortin (1979) err in their estimates of optimal depreciable lives. 5For an extended discussion of this topic see. for example, Burnside and Panton (1960). ‘In sections 2 and 3, the analysis concentrates on new personal property depreciated under the regular (non Class Life Asset Depreciation Range) system. ‘Since 195X the adjusted cost is equal to the accounting cost less additional first year depreciation (AFYD) of 20”/;, of the cost of personal property with a life of at least six years (although the IRS limits the additional first year depreciation to $2000). Since 1962, a reduction in salvage value of up to IO::, of the cost of assets with lives greater than or equal to 3 years has been allowed. These points are analysed in section 4.
Wakeman, Optimal tar depreciation
L Macdonald
215
providing an even pattern of annual tax savings over the life of the asset. Thereafter, the double declining balance’ and the sum of years-digits methods could be used for new personal property. The resulting choice of time patterns for the tax savings from depreciation is il!ustrated in fig. 1.
2c
.
-T-
7 Straight
ZC(l-s) T+1
; \’
. ~
\ ‘> t\ ‘\
Line
(dsL)
Sum of
Years-Digits
Double
Declining
(dtSYD)
-_-
(dtDDB)
Balance
_____
\ ‘\
’ \
\ \
\
\
\
\
\
\
\ x
l
\
L-a
\
--_--_
Annual Tax Savings
\ \ \ \
I Depreciation
Note: Notation:
in
Year
The area under all three curves savings from normal depreciation, C
:
Adjusted
s
:
Ratio
T
:
Allowed
T
: Marginal
*
:
Switch
is
equal to the total C(l-S)T
Cost of the Depreciable
of Allowed Useful
Salvage
Life
c
T
t
tax
Asset
Value to Adjusted
Cost
of the Asset
Tax Rate to Straight
Line Depreciation
Fig. 1. Annual tax savings from the three baw
depreciation
methods
‘After 1962, taxpayers were allowed to switch from the double declining balance method to the straight line method during the life of the asset without the permission of the Internal Revenue Service. This advantageous switch is incorporated in the analysis.
216
L. Mucdonuld
Wakernun, Optimul
tax depreciation
The set of regular depreciation methods was further widened in 1967 and 1974 to allow a taxpayer to start with the double declining balance method and switch to the sum of years--digits method or to start with the sum of years digits method and then switch to either the double declining balance or straight line methods.” The assumptions binderwhich the choice between these methods is analysed in section 3 are:
(i) The asset is either new personal
property, new residential rental section 1250 property, or section 1245 property. and is depreciated as a separate unit.“’ (ii) The sal\,age value, S. and the useful life, T. of the asset allowed in the Internal Revenue Code are given, and the asset is held for this life. (Earlier sale will not change the conclusions reached in section 3. but does affect the choice of optimal life discussed in section 4.) (iii) The appropriate tax rate is considered constant over the life of the asset. (A tax decrease will not affect the conclusions. If ;I reasonably large tax increase is anticipated the optimal strategy may be to start with the straight line method and switch to sum of years digits when the tax increase is enacted.)’ ’ (iv) The income generated each year is sufficient to cover the depreciation charged (the conclusions will still hold under the weaker condition that the ‘carry back’ and ‘carry forward’ provisions of the Code ensure that no depreciation-generated tax savings are lost, although the gains quoted herein will be smaller.)‘”
3. The optimal regular depreciation method Rather than exhaustively compare the preaent values of the tax savings generated by the depreciation methods, this section uses Descartes‘ rule of signs and the constraint that the depreciation totals for all methods arc “See KWXW Prvccdt~rc74-i 1. section IV, paragraphs (b) and (c), respecttvely. The Class Life Asset Ikprcc~ation Range System. introduced in 1971, uses essentially the stme depreaation methods and is discussed in section 4. “‘For a rewcw of the assets covered by these delinittons. set HorvitL (1Y7) ). The Income forcca~t method of dcprectation allowed in section 167(b). and generally assoctated with televtsion and firm operations, IS not discussed. Set Levin and Adess (lY74) for dctatls Nor does and deprectatton for utiltties. this paper analyze the interrelationshtp between regulation discussed tn Linhart (lY70). “See Gagnon (1976) for a detailed analysis of thts point. This analysis is also only strictly applicable to corporations and those individuals unalcted by the tax preference provistons of the Tax Reform Act of lY76. Schwartz and Livingston (lY78) and Brogdon and Fisher (iY78) show that, for certatn individuals, the tax preference penalttcs incurred outweigh the tax delay benefits of the accelerated depreciation methods. ‘lIf current losses are so large that the ‘carry back/forward’ provtsions would bc exhausted. the minimum depreciatton possible should be taken.
L. Macdonuld
217
Wakeman, Optimal tax depreciation
equal, over equal lives, to determine which method is superior. Table 1 summarises this analysis, which first proves that the accelerated methods are then shows that these accelerated superior to straight line depreciation, methods are subsets of the switching methods allowed by the 1974 procedure, and finally proves the superiority of the double declining balance switching to sum of years-digits method.
3.1. The choice
het~ren
uccclerated
Theorem 1. The uccelerutrd un_y p0sitit.e dixount rute.
methods
methods
dominate”
the strtright
Line method jkr
Proof: Consider first the sum of years-digits method. From fig, 1 we can see that the annual difference in depreciation, ~(d;“‘~-dS~), is first positive and then negative.ls By Descartes’ rule of signs, since there is only one change of sign there is at most one real discount rate, r*, which equates the present value of tax savings of the two methods.” But since the total depreciation for the two methods is equal, we know that r=O’),, is a valid solution. Hence zero is the only solution. For any positive discount rate (and for any salvage value), the present value of the positive incremental cash flows in the first half of the asset’s life will exceed that of the negative cash flows in the second half, and therefore the accelerated method dominates.16
Now consider the double declining balance method. The general pattern of incremental tax savings for (dffiB -dTL) is essentially similar to that for (dfYD-dfL), and the proof is identical. Theorem
2.
The
yeurs-digits c&e
choice
methods
depends
for
between
the
an usset
with
only upon the uppropriate
‘“In this paper, ‘A dominates B’ means net present value of tax savings generated positive discount rates. 14From
double
appendix
declining
a given discount
balunce
depreciable
und sum
rate.
that, under the assumptions set out in section 2, the by method A exceeds those from method B for all
1, we can show that
(dfYD-d~~)~C(1-s)~(T+1-2r)/T(T+1). “For
the internal
lbFurther.
equation:
CT=,
T-((d~‘D-d~L)/(l
it is simple to show that the difference ppo
is positive
rate of return
-pVSL=5.(C(1-ss)iT(T+1)).
for any positive term structure
of
/lye und salcuge
of discount
in present
i
,=I rates
+r)‘)-0.
value of tax savings,
(T+l-2r)/
I n’ (l+r,) r=,
,
L. Macdonald
218
Wakeman, Optimal tax depreciation
Proof The general pattern of the incremental tax savings is as shown in fig. 2.” Since there are two sign reversals in the incremental tax savings pattern, there are two roots that satisfy l/(l+r)~O. Since zero is one of these
Incremental Annual Tax Savings
/
1
/
// I / I /
+ T(dt
DDB _ ,s,,
\
/ I
\
)
Year
t
T
/
\ / \
/ \ \ \
Fig. 2. Incremental
“From
--
1’
tax savings between the double declining balance and sum of years-digits depreciation methods (for the case of zero salvage value).
appcndlx
1, we can show that
t=1,2 . .._.k,
(dFD” -
where k is the year for which the following condition
first holds:
L.Macdonald
Wukemann, Optimal Table
Relationships
between
the
present
Sum of Years-Digits switching to Double Declining Balance
i
$YD t
value
219
tax depreciation
I
of tax savings methods.
of alternative
regular
Declining Balance switching to SUTIof Years-Digits
Double
<
dDDB + SYD t
+ DDB
PI
v
Sum of Years-Digits
Sum of Years-Digits switching to Straight Line
I
Double
z
$.YD t
V
depreciation
Declining
Balance
dDDB t
function
of:
asset salvage discount
life value rate V
>
Straight
Line
dsL t
solutions, there can only be one positive discount rate P at which, for a given salvage and depreciable life, the present values of the tax savings from the double declining balance and sum of years-digits methodsI are equal.
The ioci of r* are presented in fig. 3 for various ratios of the salvage value to the cost of the asset, s. For a given salvage ratio, the sum of years-digits “When there present vahres is asset life, salvage values should be
are two or more sign reversals in the pattern of cash flows, the difference in not independent of the term structure of discount rates. For combinations of value and term structure which lie close to the loci shown in fig. 3, the present compared.
220
L. Macdonald Wakeman, Optimal tax depreciation
method dominates at lower positive discount rates, and at a higher discount rates the double declining balance method dominates. Given the asymmetric treatment of salvage value in the two methods, increasing the salvage value expands the range over which the double declining balance dominates (the
,s
_---
-
= 0
/-
DDB /
SYD
/ / .6
Discount Rate
I .5 I .4
.1
I
DDB
a---
’ II’
I
-
__-__-_-----
DDB ----_ SYD
1
10
5 = 0.05
20
s = 0.10
I
I
30
40 Asset
Life
* T
NOTES s:
Ratio of adjusted
the salvage cost of the
DDB:
The double declining in this region.
SYD:
The sum of years-digits this region.
value asset
to
balance
method
the
method
dominates
dominates
in
Fig. 3. The discount rate that equates the present value of tax savings of the double drclinmg balance and sum of years digits methods as a function of asset hfe and ratio of salvage value to adjusted asset cost.
loci in fig. 3 move to the southeast as the salvage ratio increases) until at relatively high salvage value/adjusted cost ratios (s~O.15), there is only one sign reversal in the pattern of incremental annual tax savings and the double declining balance method dominates for all discount rates.” 19For earlier discussions of this optimal choice problem, see Davidson and Drake (1961, 1964) and Schoomer (1966). For many countries [see ‘Principle 68’ in Price Waterhouse (1975)] these are the most accelerated methods available and therefore this is the appropriate decision.
L. Macdonald
3.2.
The choice
between
Wakeman, Optimal
switching
tax
221
depreciation
methods
Consider first starting with the sum of years-digits method and switching to either the straight line or double declining balance method. Theorem 3. Never struight line method.
switch
j’rorn
the sunl qf‘ yeclrs-digits
method
then
to the
Proof
Switching from the sum of years-digits method to the straight line method at the end of any year k (where k < T- 1) entails forgoing a tax saving now in order to gain an equal tax saving in the future.20 Thus for positive discount rates it will never be advantageous to switch. the sum of Fears--digits method is initiuted. sbtitch to rhc balance method ut the end of‘thut yeur k rjlhich is the lrqcr retllroot of the polynomial,
Theorem
double
4.
[f
declining
k2(l-s)-k(l+T)(l-s)+T(t+sT)=O. Proqf: The depreciation given by”’
while the depreciation given by
Setting
in year k+ 1, after switching
under
these depreciation
zoFrom
appendices
1 nnd
dSYD*SI._dSID=
I+ l.I
l+’
the sum of years-digits
flows to be equal
produces
at the end of year k. is
method
equation
in year k+ I is
(I), with two
2, we can show that
C(l-.\) -~-~~T(Tfl)
.(k+l-T)
for all
k
“Although the swatch to the double declining balance method allows the salvage be incorporated into the depreciable base. section 5.04 of the 1074 lnternal Revenue that:
value, S, to Code states
‘For changes in method of depreciatton to the declimng balance method, the rate of depreciation applicable to the property shall be based on the useful life of such property from the date of acquisttion, and not the expected remaining life from the date the change becomes effective’, and therefore
the depreciation
rate is 2/T rather
than 2/(T - k).
L. Macdonald
222
Wakeman,
Optimal
tax depreciation
roots, k,,,,, and kupFr. If we switch when k,,,,, $ k< kupperr the pattern of incremental tax savings (dSYD*DDB- dsYD) for t = k+ 1 to T will be first negative and then positive and therefore it will not be advantageous to switch. But if we switch when k = k,,,,Fr, the incremental cash flows will be maximising the present value of the first positive and then negative, remaining tax savings.22 Table 2 presents the values of k~,___~which maximise the present value of tax savings using the sum of years-digits++double declining balance depreciation method for various combinations of depreciable life and salvage value to adjusted cost ratio.23 Table
2
The year at the end of which a switch should be made from the sum of yearsdigits the double declining balance method.” Salvage value to adjusted cost ratio (s)
Depreciable 7
x
9
10
15
20
25
30
40
0
N
N N N D D
N N N 6 D
N N N 7 D
N
N N 18 15 D
N N 22 19 D
N N 27 23 D
N 3x 35 31 D
0.05 0.1 0.15 0.2
N N D D
life (T)
method
to
______
N
13 11 D
“N: Do not switch; the SYD method dominates the SYD+DDB D: Complex roots to the polynomial in k; the SYD-DDB DDB method (see fig. 3).
Now consider starting with the double switching to the sum of years-digits method:
declining
method. method IS dominated
balance
method
by the
and
Theorem 5. If the double declining balance method is initiuted, switch from the double declining balance method to the sum years--digits method at the end of the first year k for which
of
Proof: Although there is no gain in switching in the very early years (see fig. l), once the condition that tax savings are equal in the year following a “There is no point considering k > k,,ppcI since the present v&e will be reduced by Similarly, the optimal switching time when 05 k < k,,,,, is k=O. But this is equivalent to with the double declining balance method rather than the sum of years-digits method. 23This method is the optimal choice for assets acquired in the first half of the year if depreciated under the modified half year convention of the Asset Depreciation Range See section 4 for details.
waiting. starting they are System.
L. Macdonald
switch is met, then the declining balance-tsum methods for the remainder
Wakeman, Optimal
tax depreciation
223
differencez4 in tax savings between the double of years-digits and double declining balance of the project’s life., 2(1’1- 1 -t) ____~____ ~~~~~ (T-k)(T+l-k)
-c
t>li,
can generally25 be shown to be first positive and then negative, as illustrated in fig. 4. Therefore, the present value of the remaining tax savings will be
+
-
Fig. 4. Annual difference in tax savings between the double declinmg balance-sum digits and double declining balance methods, when the switch from double declining sum of yearsdigits is made at the end of year h.
of years balance to
‘JAlthough the switch to the sum of yearsdigtts. which takes place before a switch from the double declining balance method to the stratght line method. requires that the salvage value. \C. be excluded from the depreciable base, section 5.03 of the 1974 Internal Revenue Code states that ‘the rate of deprectation shall be based on the remammg useful life of such property’. and therefore the depreciation rate is 2/‘(T+ 1 -k) rather than 2(T - b),:? (T + I ). See appendtx 2 for detai!s. For an alternative view of the appropriate remaining useful hfe, see the discussion of the appropriate Belgian law m Gagnon and Broquet (1975). 25For non-zero salvage values, the pattern of cash flows can be first posittve. then negattve, and finally positive. See the note to table 3 for details.
224
L. Macdonald
Wakeman, Optimal
tnx depreciation
maximised if the switch from the double declining balance method to the sum of yearsdigits method takes place at the end of the first year for which @f:;S’D
Rearranging
> dDDB zzz k+,’
this equation
gives us. the condition
stated in Theorem
5.
Table 3 presents the values of the switch year, k, which maximises present value of tax savings using the double declining balance-+sum years-digits depreciation method for various combinations of depreciable and salvage value to adjusted cost ratio.26 Table The year
3
at the end of which a switch should be made method to the sum of yearsdigits
Salvage value to adjusted cost ratio(s) 0
0.05 0.1 0.15 0.2
Depreciable
life
the of life
from the double method.”
declining
balance
(T)
7
8
9
10
15
20
25
lb 2 3 N N
1
1 2 3 N N
1 2 3 N N
1 2 4 N N
1
2 3 N N
3 5 8/N’ N
1 3 5 9 N
30 1 3 6 10 N
40
1 4 7 13 N
“N: Do not switch; the DDB method dominates the DDB+SYD method. %ince the depreciation in the second year is equal under the DDB-SYD and DDB methods for zero salvage value, changing after the second year will have the same effect. ‘For this case, (d!‘Ds-sy“-dfYD) after the switch year is first positive, then negative and then again positive in years 19 and 20, i.e., there is a real root other than zero. For after-tax discount rates greater than or equal to 5.95 76, the DDB method dominates.
Having shown that all other depreciation methods are either inferior to, or subsets of the double declining balance-+sum of yearsdigits and sum of yearsdigits-+double declining balance methods, we have now only to compare these methods. Theorem 6. The double declining balance+sum of years-digits method dominates the sum of years-digits-tdouble declining balance method for all salvage ratios, all depreciable lives and all positive discount rates.
Proof: Consider yearsdigits-+double
first the case of zero salvage value. In this case the sum of declining balance method reverts to the simple sum of
Z6For assets under the half year convention, the values for k in table 3 should be Interpreted as the ji// year at the end of which the switch should take place. See appendix 4 for details.
Wakeman, Optimal tax depreciation
L. Macdonald
years-digits
method
(see table 2) and the annual 2(T+
df;D-DDB=dSYD=C.
I
1 -t)
?r(T+l)
depreciation t=l,2
’
225
is given by
,..., T.
The switch from the double declining balance method to the sum of yearsdigits method takes place at the end of year 1 (see table 3) and the annual depreciation is therefore given by dtq’B-=‘D = c
f
t=l,
t = 2,3,. . ., T
The incremental 2
(dtTB’syD - dfyD) is positive
depreciation
for the first year,
2
T’E7 and negative
thereafter
since
.2(T+l-t)<2(7-+l-t) (T-
1)T
T(T+l)
’
t = 2,3,. . ., T.
We can then conclude, using Descartes’ rule and the constraint on the total depreciation taken, that the double declining balance+sum of years-digits method dominates the sum of years-digits method for all positive discount rates. Consider now the case of positive salvage value. As the salvage value increases, the tax savings from the double declining method are unaffected in the earlier years.” However, since the depreciation base for the sum of years4igits method is reduced by the salvage value, the tax savings from switching to the sum of years-digits method are reduced, and the switch is increasingly delayed until eventually no switch takes place (see table 3). Similarly, the tax savings using the sum of yearsdigits-tdouble declining balance method are reduced in the earlier years and the switch to the double declining balance method is advanced (see table 2). The pattern of the incremental depreciation (df)DB’syD- dFrDADDR) is entirely similar to that in the zero salvage value case, although the absolute magnitude of these incremental flows is naturally larger and the initial period of positive tax savings is longer. We can therefore draw the same conclusion. “The method.
salvage
value is not deducted
from the adjusted
cost for the double
declining
balance
L. Macdonald Wakeman, Optimal tax depriation
226
To summarize
the analysis
to this point:
The optimal regular depreciation method for tax purposes is to start with the double declining balance method and to switch to the sum of years-digits method at the end of the year given in table 3.
3.3. The costs of incorrect
depreciation
choices
There are costs involved in implementing the double declining balance -*sum of years-digits method.‘* However, the consequences of an incorrect choice are by no means negligible: the use of the double declining balance method for a project costing two million dollars with a thirty year depreciable life, zero salvage value and a 5 % after tax discount rate would result in a present value of tax savings of $592,02929 compared to the $629,885 resulting if the double declining balance-tsum of years-digits method had been used - an entirely unnecessary loss of over 130/, of the initial investment. As can be seen in fig. 5a, when the straight line method is chosen the losses involved are rather large, increasing with increases in the discount rate, salvage ratio and depreciable life. The sum of yearsdigits method, which before 1974 was optimal for an asset with zero salvage value and a long depreciable life, is still relatively efficient for this case, but as can be seen in fig. Sb, the losses increase substantially with increases in the salvage ratio. In contrast, the losses incurred when the double declining balance method is used are quite large for long-lived assets with zero salvage value (see fig. 5c), but decrease with increases in the salvage ratio - to the point where, for salvage ratios exceeding 15 %, the simple double declining balance method is optimaL3’ The conclusion of this section is that the administrative costs of implementing the double declining balance+sum of years-digits methods would have to be very high indeed to outweigh the tax benefits of this method. 28Rev. Proc. 74-77 states: ‘Change in method of depreciation. Taxpayers may expeditiously obtain, under the prescribed conditions, the Commissioner’s consent to change certain methods of depreciation accounting to certain other methods by filing application on Form 3115, within the first 180 days of the taxable year the change is to become effective, with the Director of the Internal Revenue Service Center where the return will be tiled.’ Clyde Stickney notes that not only must notice be given to the IRS, but detailed information about acquisition cost, cost additions, depreciation taken in prior years, salvage value, and other items must be supplied for each asset. “Assuming
a 50% tax rate.
30Appendix 3 presents calculations of the present value of tax savings methods for various value of asset life, discount rate and salvage ratio.
of the alternative
L. Macdonald Wakeman, Optimal tax depreciation
r
50%
(a)
Straight
Line
221
Depreciation
Percentage Loss
Asset
(b) Percentage Loss
Life
Sum of Years-Digits
S = .lO
1
25%
--
___---
5
10
15
20
25
Asset 50%
_-
__--
*.________---
O%-
HA
/
r (c)
Double
30
35
s
= .os
=
0
S
=
0
s
se
.05
s =
.lO
40
Life
Declining
Balance
Percentage Loss
___
5
Note:
s = ratio
10
of
15
salvage
20
--_
----
25
30
Asset
Life
value
to
35
adjusted
40
cost
Fig. 5. Reduction in the present value of tax savings incurred by an incorrect choice of depreciation method, as a percentage of the present value of the double declining balance-tsum of years-digits method; tax rate = 50 %, discount rate = 10%.
228
L. Macdonald Wakeman, Optimal tax depreciation
4. Extensions The above analysis can also be applied to the choice of depreciation life and to analyse the Class Life Asset Depreciation (ADR) System.
4.1. The investment
optimal Range
tax credit
Under the investment tax credit programme, business firms can deduct, as that year. a credit against income tax, a percentage31 of this new investment For the full loo/:, credit to be taken, an asset must have a depreciable life of at least seven years. For assets with lives of live to six years, the credit is reduced to 0 of lo%, and assets with three to four year lives receive a credit of only 3 of loo/,. The question ‘What is the optimal econorkc life to choose?’ is beyond the scope of this paper since other cash flows, such as maintenance, will no longer be constant when comparing unequal-lived assets. However, we can determine which depreciable life3’ should be considered for each economic life. If the IRS allows depreciable lives shorter than the given economic life to be chosen, the appropriate cash flows per investment dollar are those given in table 4. Since the total cash flows per investment dollar are equal for the 3 and 4 year depreciable lives (i.e., 0.51333) and for the 5 and 6 year depreciable lives (i.e., 0.54667) and the sign of the incremental cash flows (e.g., 3 years minus 4 years) changes only once we can immediately conclude that the shorter lives dominate for all positive discount rates. ‘For the comparisons in which the total cash flows differ the incremental cash flows still change signs only once,33 and therefore we know that there can be only one positive discount rate which equates the net present values for the different depreciable lives. As shown in table 5, the longer life is optimal for all lower discount rates, and for any higher discount rate, the shorter life is preferred.3” The analysis for depreciable lives longer than the given economic life differs slightly since in this case the depreciation for the years beyond the asset’s economic life is taken at the end of the asset’s last year and the investment tax credit ‘excess’ will be recaptured when the machine is written “initially, in 1962, the percentage was 7 “i,. Under the current (1975) legislation, the credit is lo%, with an extra l-1$% being allowed for firms which meet certain criteria with respect to their employee stock option plans. 3ZTo the extent that the IRS permits. “Except the comparison of the 6 and 7 year depreciation lives for a 7 year economic life. The incremental cash flows in this case change sign twice, but we can show that rcr0 for one of the solutions. 34F6r example, an asset with an economic life of 5 years should be depreciated over a 5 year life if the discount rate is 11.4% or lower and over a 3 year life if the discount rate is higher.
L. Macdonald
Wakeman, Optimal Table
The after-tax”
cash flows per investment
Depreciable
dollar
229
tax depreciation
4
from depreciationb
and the investment
tax credit.
life 5
6
1
0.25867 (20/50r + Z/30)
0.22667 (151457 + 2/30)
0.23714 (421147~ + 3130)
0.12 (3/12~)
0.1152 (121507)
0.10667 (10/457)
0.09796 (30/1477)
008 (21127)
0.0864 (9!5or)
0.08533 (R/457)
0.08163 (25/147~)
0.04 (1112r)
0.0576 (6!5&)
0.064 (6/45r)
0.06531 (20/147r)
0.0288 (3/5Qr)
0.04267 (4/45r)
0.04898 (15/1477)
0.02133 (2/45~)
0.03265 (10/147T)
Year
3
4
I
0.35333’ (6/9r+ 11’30)
0.27333 (6/12r+
2
0.10667 (2197)
3
0.05333 (1197)
4
1,‘30)
5 6
0.01633 (5/147?)
I Total cash flow
0.51333 (7 + l/30)
0.51333 (T + l/30)
0.58 (T + 3130)
0.54667 (r.22130)
0.54667 (T + 2/30)
“A tax rate, T, of 48 y, is used bThe double declining balance *sum of yearsdigIts method is used. For this calculation, a zero salvage value is assumed. ‘For a 3 year depreciable life, the investment tax credit per investment dollar is f (lo%), and the first year depreciation under the double declining balance method is 2(1/3)r per investment dollar, for a total tax credit of 6/9t + l/30. For a tax rate of 48 %, this credit is worth 0.35333 dollars per investment dollar. Table The optimal
Economic life
depreciable
life for a given economic
Depreciable -
5 life as a function
of after-tax
discount
rate r.’
life 3
4
5
6
7 D
LL
r>O
D
D
D
SL LL
r>O -
D r>O
-
-
-
D
D
D
SL LL
r> 11.47” -
D -
rs 11.4’;b r>O
D
SL LL
r>11.4”,, _
D _
r<11.49/, -
D D
r>O
SL
r>12.7%
D
D
D
rs 12.7 %
_ D
“SL: If the IRS permits a depreciable life shorter than or equal to the planned economic life. LL: If the IRS only permits a depreciable life equal to or greater than the planned economic life. D: Dominated solution.
230
L. Macdonald
Wakeman, Optimal tax depreciation
OfF.3s Since there are no differences in the investment tax credit taken, the total cash flows are again equal for all comparisons. There is still only one incremental cash flow sign change36 and therefore the shorter life dominates3’ If for a given economic life any depreciable life is permissible, the solution is the same as that presented in table 5 for the shorter life option. Additional first year depreciation (AFYD) allows a taxpayer to shift up to 209; of the total depreciation (up to a maximum of $2,000 in a given year) from the last years to the first year3* for assets with lives of at least six years. Therefore AFYD favours the 6 and 7 year depreciable lives. However, since the additional depreciation is limited to the first $10,000 of investment, the impact of this provision on the choice of optimal life is limited to assets which cost less than $100,000.39 The conclusion is that for any planned economic life, asset cost, and discount rate, there is one optimal depreciable life which quite often differs from the economic life for short-lived assets.
4.2. The class life asset depreciation
range
(ADR)
system
The foregoing analysis can also be used to choose the optimal depreciation method and optimal depreciable life for assets which are depreciated under the ADR system. The ADR system, introduced in 1970, differs from regular depreciation in several important ways : (i) Salvage value is ignored for all depreciation methods. (ii) The depreciable life can be chosen in the range of the guideline life plus or minus 205;. (iii) The set of applicable depreciation methods is larger, since there is a
35For example, when comparing 5 and 7 year depreciable lives for a 5 year economic life, and remaining depreciation is taken in year 5 and, if a 7 year depreciable life has been chosen, f of I am indebted to Mark Wolfson for clarifying this the lo”,, investment tax credit is recaptured. pomt. “Again, the comparison between the 6 and 7 year depreciation lives for a 6 year economic life differs. There are three sign changes and therefore three roots. However, the two non-zero roots are both negative. Contrary to the results for all other comparisons. the 7 year depreciatton hfe dominates the 6 year life for all posittve discount rates. 37For example, for a 4 year economic hfe. the 4 year depreciable life dominates all longer depreciable lives for any positive discount rate. 381t IS normally optimal to apply the AFYD to the longest-lived asset purchased in a given year, AFYD affects the opttmal depreciable life choice only if less than $10.000 is spent on longlived assets in the year. 39For a $100,000 Investment, the normal first year depreciation for a 7 year life is given in table 5 as $23.714. Even with the additional first year depreciation of $2,ooO the total first year depreciation of $25,714 is less than the first year depreciation for a 5 year life of $25,867. Since the sign pattern is unchanged, the dominance results remain unchanged.
L. Macdoncrld
Wakeman, Optimal
231
tax depreciation
choice of first year conventions - the .half year convention the modified half year convention (MHYC).40
(HYC) and
Since Stickney and Wallace (1977), using computer simulation, survey the optimal choices of depreciable life, depreciation method and first year convention for the ADR system,4’ I will concentrate on those points where the analytical technique compliments or illuminates their simulation results. Consider first the half year convention (HYC) for ADR assets.42 The depreciation flows are equal to those taken under the normal half year convention, and since the appropriate salvage rate is zero, the optimal depreciation rule is simple: Start with double declining balance after the first full year of depreciation.
and switch
to sum of years-digits
The modified half year convention (MHYC) differs from the HYC convention in that assets acquired in the first half of the year get a full year’s depreciation in the first year while assets acquired later in the year receive no depreciation that year. The depreciation flows for assets acquired in the second half of the year under the MHYC are similar to those for the HYC, and lead to the same optimal depreciation rule: Start with double declining balance the beginning of the third year.
and switch to sum of yearsdigits
at
But the combination of the full first year depreciation and the extra half year of remaining life for assets acquired in the first half of the year under the MHYC favours the sum of years-digits43 method. The result is that the depreciation taken under the double declining balance+sum of years-digits method in the year after the optimal switch is actually less than the sum of years-digits depreciation for that year. Hence the analysis developed in Theorem 6 to prove the dominance of the double declining balance-tsum of years-digits method is no longer relevant. Stickney and Wallace (1975) go so far as to state that ‘the sum of yearsdigits method is preferable for all depreciable lives greater than four’. Analytically, this statement is incorrect since there are two changes in sign of the incremental depreciation, (@DB-sYD -,fYD), implying that there is at least one real discount rate, r*, other than zero, for which the net present values of the two methods are equal. The locus of these rates for differing asset lives has a shape which is similar to the s=O locus in fig. 3. However, r* is 16.1 “/o, ‘“For further details of the Class Life (ADR) System. see Feinschreiber (1975) or Stickney and Wallace (1977). “See especially their tables A-l to A-24 and B-l to B-22. “‘Assets receive a half year’s depreciation in the year of acquisition, and remaining life is set equal to (7--j ). See appendix 4 for details. “The depreciation m year r under the sum of years+digits method is now 2(T+ l.S- t)/ T(T+l)instead of2(T+l-t)/T(T+l)for t>l.
232
L. Mncdonald
Wakeman, Optimal
tax depreciation
41.9 ?,, and 59.6 y0 for the 4, 5 and 6 year asset lives respectively, and since Stickney and Wallace, quite reasonably, used a maximum discount rate of 20 “,, in their simulation, it is easy to see why they made the error, For practical purposes, their rule is optimal for assets acquired in the first half of the year under the MYFC with less than loo/;, salvage value.44 Since the analysis of the impact of the investment tax credit under the ADR system is similar to that described in section 4.1,45 I refer the interested reader to Stickney and Wallace for the results.46 The conclusion of this section is that analytical solutions for optimal depreciation strategies under the Class Life (ADR) System can be easily derived.
5. Conclusion
To date, solutions to the choice of optimal depreciation strategies for tax purposes have generally been found through numerical simulation. This paper presents a simple capital budgeting technique which is used to derive analytical optimal depreciation strategies4’ and which will probably be equally useful in analysing future changes in the Tax Code.
Appendix 1: Simple depreciation methods
The following C: S: s: T: A,:
notation
is common
to all methods:
Adjusted cost of the depreciable asset, Allowed salvage value of the asset, s/c7 Allowed useful life of the asset, Depreciable base of the asset at the end of year r, where t=O, 1,. . ., T.
44Although salvage value does not affect the depreciable base for the SYD method, it does change r* smcc it differentially affects the timing of cash flows in the last years. For example, if a salvage coefficient of .s=O.l is applied to the 5 year life, the depreciations under the double declining balance-sum of years-digits method terminates in year 4 rather than year 6 while the sum of years-digits method depreciation still terminates in year 4. Then the double declining balance-sum of years-digits method dominates the sum of years-digits method for all discount rates of 1‘I;, or more. The general effect of salvage value on the choice of optimal depreciation method for the MFYC is similar to the effect of salvage value on the choice between double declining balance and sum of yearsdigits illustrated in fig. 3. “Although the range of depreciable lives for a given guide-line life is limited to ~20’;,, around that guide line life. Mark Wolfson points out that there is a potential for even more flexibihty since asset classifications are not iron-clad. “For example, their Table A-2 shows that, under the HYC, an asset with a gmde-line of 4 years should be depreciate - over 5 years for discount rates of 12”;, or less and over 3 years for discount rates of 13 :‘., or higher. “‘With the bonus of explaining why a certain depreciation strategy is optimal under the given clrcurnstances.
L. Macdonald Wakeman, Optimal tax depreciation
A-l .I. Straight
233
line depreriution
The initial depreciable Salvage Value, AEL=C(l
base
equals
the Adjusted
less the Allowed
-s). taken in year t under the straight
The depreciation
Cost
line method
is therefore
and AFL by A;L=C(l
A-l 2.
Sum
-$$!,
qf yews-digits
t==O,l,...,
7‘.
depreciation
A;YD=C(l
-s),
dsy”=C(l t
3(T+ -.s).-rcr+F,
and
A;”
A-1.3.
Double
1 -t)
(T-thT+ = C ( I- s ) m--(T+r-.
declining
balance
t-l,.?,...,
1 -t)
T
t=O,l,...,T
depreciation
A switch from the double declining balance method to the straight line method is allowed, and takes place at the end of the first year, k, for which the tax saving in the following year from switching to straight line is greater than that from staying with double declining balance. Before the switch,
L. Macdonald Wakeman, Optimal tax depreciation
234
The switch is made condition first hoi&: dDDB’SL f,k
at the end
> dDDB =t
of the year
k for which
the following
9
i.e.,
After the switch,
Appendix 2: Switching between accelerated methods A-2.1. Double declining
balance to sum of years-digits
The depreciation taken under switch from the double declining given by
the sum of years-digits method after a balance method at the end of year k is
2(T-k+l-t) ~T_k~~T_k+l~~
-
t=LT...,(T-kL
since under section 5.03 the remaining life of (T-k) is the relevant depreciable life, and salvage value must be deducted to get the depreciable base for sum of years-digits. Thus,
t=k+l,
k+2 ,..., T.
235
L. Macdonald Wakeman, Optimal tax depreciation
(The problem of a switch from double declining balance to straight line before a switch from double declining balance to sum of years-digits can be disregarded since the tax savings in the year after the switch, k + 1, will be greater for a switch to sum of years-digits than for a switch to straight line).
A-2.2. Sum of years-digits to double declining balance The depreciation taken under the double switch from the sum of yearsdigits method d;;D-DDB = C (1 -s)
declining balance method after a at the end of year k is given by
(T-k)(7-+1-k)+s T(T+
1)
t=k+l,k+2
,..., T.
Appendix 3: The net present value of alternative depreciation methods Throughout this appendix, it is assumed that the initial value of the asset is $1,000,000. After-tax Present Values can be obtained by multiplying the figures presented by the appropriate tax rate. Table A.1 Present value of depreciation stream; salvage value/cost ratio = 0.0 (initial value of asset: rsl,OOO,OOo). Life of asset
Discount rate
SL
SYD
DDB-SL
DDB+SYD
5
0.05 0.10 0.15
865895 758158 67043 1
89403 1 806142 132315
896505 810998 739427
899866 816437 746094
10
0.05 0.10 0.15
772174 614457 501877
828460 700988 603785
816919 685282 587557
831214 705612 609692
20
0.05 0.10 0.15
623111 425678 312966
717885 546973 436211
68925i 515387 408780
719119 548879 438492
30
0.05 0.10 0.15
512415 314231 218866
629142 442432 335971
592029 409107 311278
629885 443505 337198
40
0.05 0.10 0.15
428977 244477 166044
557096 368548 271205
516634 338254 251379
557602 369241 271972
L. Macdonald Wakeman, Optimal tax depreciation
236
Table stream;
A.2
Present
value of depreciation
salvage value/cost rsl,OOO,c0O).
ratio=O.l
(initial
value of asset:
Life of asset
Discount rate
SL
SYD
DDB+SL.
DDB+SYD
5
0.05 0.10 0.15
779306 682342 603388
804628 725528 659138
817306 747564 688099
817306 747564 688099
10
0.05 0.10 0.15
694956 553011 451689
745614 630889 543407
751793 641912 558085
754694 645866 562209
20
0.05 0.10 0.15
560800 383111 281670
646096 492276 392590
644747 494575 398603
654166 504331 406567
30
0.05 0.10 0.15
461174 282808 196919
566228 298189 302374
561529 398994 301665
514279 40929 1 314441
40
0.05 0.10 0.15
386080 220029 149440
5Ot 386 331698 244084
495659 333274 250060
509527 342119 354934
Appendix 4: The regular half year convention The regular half year convention considers all acquisitions to be made at the middle of the taxable year. Thus, the depreciation taken in the first half year is 5 of the first year’s depreciation, that taken in the second year (first full year) is ) of the first year plus ) of the second year, etc. The condition for a switch from double declining balance to sum of yearsdigits at the end of the full year (k=O, 1,2,. . ., T) is
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L. Macdonald
Wakeman,
Optimal
tax depreciution
231
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