The Effects of Depreciation Allowances on the Stochastic Replacement Decision

1
The Effects of Depreciation Allowances on the
Stochastic Replacement Decision
by Roger Adkins and Dean Paxson
Correspondence r.adkins_at_salford.ac.uk
2
Depreciation tax shield as an investment
incentivising device
Deterministic Investment Model based on NPV
Brown (1948), Domar (1953), Goode (1955),
Barritt (1959), Davidson Drake (1961, 1964),
Schoomer (1966), Wakeman (1980)
Accelerated depreciation preferred to straight
line
Stochastic Investment Model based on Dynamic
Programming
Berg Moore (1989), Berg et al. (2001),
Wielhouwer et al. (2002), De Waegenaere
Wielhouwer (2002)
Position is mixed
Real Option Stochastic Replacement Model
Mauer Ott (1995) also Ye (1990), Dobbs (2004)
One factor model by treating depreciation as
function of operating cost Compromise
3
Aims
To develop a quasi-analytical solution to the
two-factor stochastic real option replacement
model with operating cost and depreciation
1.
Brennan Schwartz (1985) claim that tractable
analytical solutions do not exist
Quasi-analytic
Solution to set of simultaneous non-linear
equations (not closed form)
Easier solution interpretation than finite
difference method
Not vulnerable to accumulating errors of finite
difference method
2.
To evaluate the effect of tax effects through the
depreciation schedule on replacement decision
Perspectives of users, suppliers and tax
authorities
Compare the effects of controllable parameters on
decision
4
Outline Stochastic Real Option Replacement Model
Declining Balance (DB) Depreciation Schedule
Model Interpretation
1.
Straight Line (SL) Depreciation Schedule Model
Interpretation
2.
Comparison of DB with SL
3.
Focus on interpreting the results
Complete explanation of the analytical
development supplied in paper
Assumptions
Absence of competition, other option
opportunities irrelevant, fixed replacement
investment cost, only input decay relevant
(constant revenues) with operating costs
following gBm, strictly proportional taxation
5
Declining Balance Depreciation Schedule
Depreciation D
gBm with zero volatility
Ito Lemma Derive two-factor risk neutral
valuation relationship (RNVR) as pde Solve
two-factor pde
Time T
Characteristic roots of RNVR
1.
Discriminatory boundary as a mapping of the
trigger levels for operating cost and depreciation
Value matching relationship
2.
Smooth pasting conditions
3.
Express discriminatory boundary as function of
asset age since last replacement
Initial depreciation level at replacement
6
Simulation Data for DB Replacement Model
Average asset lifetime based on proportional
depreciation rate
Entire depreciation cost is depreciable
Constant revenue level is irrelevant
7
Discriminatory Boundary for DB Replacement Model
Operating Cost Trigger Level
Mauer Ott
Dobbs
Asymptotic Limit
B
Replacement Region
Continuance Region Below AB
Present Formulation
A
Asset Age
Replacement policy depends on asset age
8
Effect of Operating Cost Volatility Changes
Operating Cost Trigger Level
Value for Asset with Embedded Option
Operating Cost Volatility,
Replace at greater operating cost in presence of
volatility
Deterministic Solution
9
Effect of Operating Cost Volatility Change on
Boundary
Operating Cost Trigger Level
Asset Age
Replacement at lower operating cost for lower
volatility for all asset ages
10
Effect of Average Depreciation Lifetime (ADL)
Change on Boundary
Operating Cost Trigger Level
Preference ranking is mixed, but
Least ADL preferred for young or old assets
Greatest ADL preferred for assets with in-between
ages
Asset Age
11
Effect of Corporate Tax Rate Change on Boundary
Operating Cost Trigger Level
Lowering tax rate only favours lower operating
cost for older assets
Asset Age
12
Effect of Initial Operating Cost Change on
Boundary
Operating Cost Trigger Level
Lowering initial operating cost of replica
unambiguously leads to lower operating cost
trigger
Asset Age
13
Effect of Replacement Investment Cost Change on
Boundary
Operating Cost Trigger Level
Lowering replacement investment cost of replica
unambiguously leads to lower operating cost
trigger
Asset Age
14
Effect of Initial capital Allowance Change on
Boundary
Operating Cost Trigger Level
Asset Age
Raising initial capital allowance unambiguously
leads to lower operating cost trigger
15
Straight Line Depreciation Schedule
Remaining Cumulative Depreciation X
Initial cumulative depreciation
aBm with zero volatility
Depreciation lifetime
Ito Lemma Derive two-factor risk neutral
valuation relationship (RNVR) as pde Solve
two-factor pde
Time T
Characteristic roots of RNVR
1.
Discriminatory boundary as a mapping of the
trigger levels for operating cost and depreciation
Value matching relationship
2.
Smooth pasting conditions
3.
Express discriminatory boundary as function of
asset age since last replacement
for
16
Additional Simulation Data for SL Replacement
Model
Entire depreciation cost is depreciable
Average depreciation lifetime
Both the DB and SL replacement models have
identical average depreciation lifetime
17
Discriminatory Boundary for SL Replacement Model
Operating Cost Trigger Level
Dobbs
Replacement Region
Present Formulation
C
B
Continuance Region below line ABC
B is point of asymmetry at N20
A
Asset Age
18
Comparison of Values for DB and SL Replacement
Decision
Value of Asset plus the Replacement Option
No universal preferred schedule
DB preferred
SL preferred
Negative since revenue omitted
DB preferred
DB
SL
Asset Age
Average depreciation lifetime
19
Conclusions for both DB and SL Replacement Models
Lowering operating cost trigger level benefits
Asset users through greater cost efficiencies, but
Replacement option value also decreases
Asset suppliers through more frequent replacements
Tax authorities through greater tax take
Economic policy through greater replacement
investments
20
Conclusions for both DB and SL Replacement Models
Change Negative Negative Negative Negative Positiv
e
Factors Operating cost volatility Operating cost
drift rate Initial operating cost
level Replacement investment cost Initial capital
allowance
Reduction in operating cost trigger level for all
asset ages Unambiguous preference ranking
Change Negative Negative
Factors Average depreciation lifetime Corporate
tax rate
Reduction in operating cost trigger level for
some asset ages Ambiguous preference ranking
Improving cost efficiency
Asset Suppliers Lower volatility, initial
operating cost and replacement cost
Tax Authorities Raise initial capital allowances
Asset users Lower average depreciation lifetime
is ambiguous
21
Comparison of DB and SL Replacement Models
Deterministic replacement model
DB universally preferred to SL
Real option stochastic model
No universally preferred depreciation method
DB is preferred for younger and older assets
SL is preferred for ages in-between
Replacement policy depends on asset age
22
Possible Extension of Quasi-Analytic Method
Straight line schedule and resource extraction
share a similar RNVR
Remaining Reserves X
Time T
The possibility of extending quasi-analytic
method to models involving assets having a finite
life
Provide an analytical solution to Brennan
Schwartz (1985)