swp 47/87 the stock depreciation model of new car sales

SWP 47/87
THE STOCK DEPRECIATION
MODEL OF
NEW CAR SALES - A RECONSIDERATION
DR FRANK F’ISHWICK
Reader in Managerial Economics
Cranfield School of Management
Cranfield Instst~e~~ Technology
Redford MK43 OAL
United Kingdom
(Tel: 0234-751122)
(Fax: 0234-751806)
Copyright: Fishwick 1987
.
1
f
\,
THE STOCK DEPRECIATIONMODELO F N E W CAR SA
- A RECONSIDERATION
Introduction
l
This research was stimulated
by a desire to see whether any
single
explanatory
model could accommodate the large and
fairly
rapid changes in demand for new passenger cars in the
countries
over
United Kingdom and other Western European
tests suggested that the
the 20 years to 1984. Preliminary
model based on "stock adjustment"
or llstock depreciation"
The origins
m ight well prove satisfactory
in this respect.
and basic features
df this model are outlined
in Section 1
of this paper.
Section 2 shows that the model remains remarkably consistent
with data for the 20 years to 1984, both for the United
in demand
Kingdom and F rance, despite the wide variations
if forecasters
had
In both these countries,
conditions.
been able to predict
changes in real consumers' expenditure
of the model
and the relative
prices of cars, application
in the early
months of one year would have provided
a
estimate
of new car registrations
in
reasonably
accurate
For West Germany and Belgium, the
that year and the next.
though the results
suggest
model proves less satisfactory,
that attempts at refinement may be justified.
-
Section 3 reports use of prices information
to analyse some
the
of the model, especially
of the basic
assumptions
assumption that the rate of depreciation
does not vary with
research leads to rejection
,of this
The empirical
age.
3
assumption and it is concluded that the apparent success of
\r
the model in explaining
new registrations
in the United
i
Kingdom and F rance may have been partly
fortuitous.
2
The Basic
1.
Model
The origins
of the stock depreciation
model are generally
It was applied to
associated
with Stone and Rowe, 1957 (7).
States
by
the demand for passenger
cars in the United
Kingdom by Dicks-Mireaux,
Nerlove
(4) and in the United
O'Herlihy
published
some revised
work
O'Herlihy
et al (2).
Further reference
relating
exclusively
to cars in 1965 (5).
to the stock depreciation
model with application
to the
United States appears in work by Smith, published
in 1975
The most recent
application,
again to the United
(6)
(1) I
Kingdom, seems to be that of Deaton and Muellbauer
who applied it to British
data for 1954-75.
l
l
The underlying
assumption of the model is that desired stock
(adjustment
for
.equivalents
(S*) measured in new vehicle
has a stable
vehicle
size is sometimes also introduced)
income
relationship
with economic variables
(for example,
The model also assumes that the ratio
between
and price).
the value of a car of any age i years and that of the same
of
model aged i - 1 is (n - 1)/n, where n is the reciprocal
(1 minus
the
annual
rate
of
depreciation),
normally
Following
Stone and Rowe (7) it
expressed as an integer.
has been further
assumed that adjustment
to desired
stock is
not achieved during the current year, so that actual stock S
= rS* where r is less than unity.
If r = 1 and x is an independent
variable
to which S* is
then new registrations
q in year t can be
linearly
related,
determined
by
St = St - (n - 1) St-1
n
=a+
X
=
Xt
bxandhx
_(n - 1) (a + b(x-bx)),
n
= Xt - xt-1
where
'
.
3
.I
This sim p lifie s
into
th e fo l l o w i n g
9 t = a ' + b 1 (x + (n - 1)
AX),
e q u a tio n :-
w h e r e a ' = a /n a n d b ' = b /n
first,
all
are required:
If r # 1 , th e n tw o correctio n s
b y r.
th e te r m s in th e e q u a tio n n e e d to b e m u ltiplied
S e c o n d l y , a n e x tra te r m n e e d s to b e a d d e d in o r d e r to ta k e
into a c c o u n t th e failure
to a d j u s t to l o n g - te r m equilibrium
S ince, in year t - 1 ,
in th e previous year.
S t-1 = r(S *tB l - S tB l)
th e a d d i tio n a l
for i n c o m p l e te
purchases
a d j u s tm e n t
r e q u i r e d in year t to c o m p e n s a te
in t - 1 a r e g i v e n b y (1-r)qtB l.
T h e e q u a tio n to b e e s tim a te d for a d e p r e c i a te d
w ith k i n d e p e n d e n t variables
is, th e r e fo r e
qt
= ra'
+ rbIl
stock
( 1 + ( n - 1 ) A )x1 + r b 1 2 ( 1 + (n-1)8
....
r&t
model
)x2 +
1 + (n-1) A )xk + (l-r)qtwl
In o r d e r to derive this e q u a tio n , it is necessary to specify
a linear relatio n s h i p
b e tw e e n d e s i r e d stock a n d e a c h o f th e
A ll a u thors q u o te d a b o v e h a v e u s e d
i n d e p e n d e n t variables.
this linear specifica tio n .
2.
P reliminarv
A.
U n ite d
T e s tin s
o n D a ta for R e c e n t Y e a r s
K insdom
It w a s d e c i d e d to te s t a fairly
sim p le m o d e l w ith only tw o
to
i n c o m e a n d th e o th e r to th e
o n e related
variables,
P reliminary
te s tin g
c o n firm e d th e
relative
price o f cars.
view received fro m analysts w ith i n th e m o tor i n d u s try,
th a t
relate d
to
w e r e m o r e closely
registra tio n s
n e w car
c o n s u m e r s ' e x p e n d i tu r e th a n to p e r s o n a l d i s p o s a b l e i n c o m e .
includes a n e l e m e n t o f n o n This m a y b e b e c a u s e th e latter
The
( m o r tg a g e r e p a y m e n ts, e tc).
discretio n a r y
"savings"
.
4
implied
price variable
used in this study was the deflator
and
expenditure
on cars
by the
figures
of consumers'
in the National
Income and Expenditure
Blue
motorcycles
Indices based on list
prices of new cars weighted by
Book.
registrations
proved unsatisfactory,
possibly
because they
did not capture the discounting
which has taken place in the
retail
market since 1980.. The derived price index for cars
was divided
by the general index of retail
prices
in order
The 'regression
results
to produce a relative
price index.
for 1964-1984 (21 years) were as follows:Coefficient
*
W
R2
(ii)
jp
I
0.942
DW = 1.36
0.942
Constant
(1+3A) CEt
(1+3A) Pt
qt-1
DW = 1.24
-392.80
+0.0145
-204.54
+0.075
qt =
where
=
=
Std.error
'of coefficient
123.64
0.0008
98.62
Constant
(1+3A)CEt
(1+3A)Pt
-375.87
+0.0155
-243.58
9t =
Variable
126.30
0.0014
109.97
0.089
CE
=
consumers' expenditure
at 1980 prices
P
=
index of prices
of retail
prices
qt
=
new registrations
thousands
in year t measured
qt-1
=
new registrations
in year t-l
in millions
of pounds
of cars and motorcycles/index
1980 = 1
in
in thousands
The matrix
of correlation
coefficients
relating
to the
variables
in equations
(i) and (ii)
appears in the Appendix.
This shows that
in equation
(i) there
is no significant
correlation
between the consumers' expenditure
and relative
there
is significant
In equation
(ii),
price
variables.
in year t and new
correlation
between consumers' expenditure
is probably
not
This
in year t-l.
car registrations
-
5
.
sufficiently
serious to distort
these independent variables.
the coefficient
The estimates
provided by equations
(i)
in Figure 1 and Figure 2 respectively.
evident:-
on either
of
and (ii)
are shown
Some features
are
(1) Both equations give a good fit over the entire
period;
are inconclusive,
although
the Durbin Watson coefficients'
this is mainly because of runs of very small residuals.
*
(2) .The value of the reciprocal
of depreciation
(n = 4,
iteratively,
is
to 25%); which was derived
equivalent
in this
This will be examined further
reasonably
plausible.
paper.
year's
(3) The regression
coefficient
on the previous
(l-r)
is
not
equivalent
to
which
is
registrations,
significantly
different
from zero and the inclusion
of this
This
variable
does not improve the regression
results.
implies
that adjustment
of S to S* is achieved within
the
year in question
and that the partial
adjustment
factor
of
Stone and Rowe is not necessary in this case.
These results
are similar
to those reported
by Deaton and
They
Muellbauer
(1) for UK data covering the years 1954-75.
(real
variables
economic
independent
used
two
also.
By iteration
(to minimise the
disposable
income and price).
standard error
of estimate)
they found a depreciation
rate
of 22 per cent (n = 4.5) compared with the 25 per cent
Evidence
from
above.
implied
in equations
(i) and (ii)
Mogridge (3) and from calculations
presented
later
in this
paper indicates
that depreciation
rates have tended to rise
year in the Deaton and Muellbauer
since
1975, the last
analysis.
.
_
6
regression
the
that
found
also
same authors
These
different
coefficient
(l-r)
on qtWl was not 'significantly
Both
from zero implying
that r might be close to unity.
O'Herlihy
1965 (5) and Smith (6) report
a similar
finding.
It
may be argued
a priori
in the case of cars
that
postponement
of purchase decisions
by some consumers may be
compensated by purchases of other consumers anticipating
an
improvement in economic circumstances.
As a test
short-term
adopted:-
%
of
the usefulness
of
the
forecasting,
equations
following
(i) and (ii)
procedure
(a>
regressions
equations
were calculated
for groups of
with 1964-78 and ending
consecutive
years, starting
with 1969-83;
(b)
it was assumed that
predict
accurately
expenditure
and the
years beyond those
in
was
15
the forecasters
were able to
the behaviour of consumers'
relative
price of cars in the two
on which regression
had been based;
.
.
(cl
in the case of equation
(i), the actual values for
in order to obtain
(1+3A)CEt and (1+3A)Pt were.inserted
forecasts
of new registrations
in the first
and second
years following
those included in the regression;
W
in the case of equation
(ii) new registrations
were
forecast
for the first
year after that covered by the
by insertion
of the actual values for
regression,
(1+3A)CEt,
(1+3A)Pt, and qtWl;
W
the result
of stage (d) was
also for equation
(ii),
used as the appropriate
value for qtW1 in a forecast
new registrations
in the second year beyond those
covered by the regression.
.
of
7
t
be made during
The forecast
made "one year earlier"
would
the first
few months of the year to which it referred,
The
because of delays in the availability
of statistics.
forecast
for'the
following
year would be made at the same
shown in Table 1, suggest that the stock
The results,
time.
provides
a satisfactory
only
not
model
depreciation
explanation
of changes in the demand for new cars during the
period 1964-84 but may also assist practical
forecasting.
Table
1
Test of stock depreciation
United Kinsdom
models in forecasting
Number of new resistrations
Actual
Year
Equation
Equation
1979
1980
1981
1982
1983
1984
_
made
2 years
earlier
(i)
1732
1536
1514
1639
1871
1764
1979
1980
1981
1982
1983
1984
Forecast
1 vear earlier
(000s)
1721
1396
1423
1540
1835
1726
1426
1377
1521
1818
1721
1737
1475
1455
1551
1840
1795
1505
1418
1526
1804
1784
(ii)
1732
1536
1514
1639
1871
1764
-
8
B.
France
Equations
equivalent
to (i) and (ii)
above for the United
the period
Kingdom were calculated
for French data covering
index was that published
by
The relative
price
1965-84.
by the general index of retail
INSEE for new cars, deflated
prices.
variable
Coefficient
(iii)
qt
=
(iv)
st
=
=
206.8
0.15
167.6
0.949
DW = 1.72
184.47
0.44
197.84
0.15
Constant
(1+3A)CEt
(1+3A)Pt
St-1
400.29
+3.06
-460.96
-0.14
R2
Std.error
of coefficient
Constant
(l+3A)CEt
(1+3A)Pt
324.65
+2.67
-363.70
fi2
.
=
0.948
where CE = consumers' expenditure
1970 prices
DW = 1.81
in billions
P
= index of prices of new cars/index
prices
- 1970 = 100
qt
= new registrations
thousands
of francs
at
of retail
in year t measured
in
The matrix
of correlation
coefficients
in equations
(iii)
It will
be seen that
and (iv)
appears in the Appendix.
there
is no significant
correlation
between
consumers'
consumers'
but
price
car
relative
expenditure
.- and
t is significantly
correlated
expenditure
in time period
This probably
with new registrations
in time period t-l.
contributes
to the reduced significance
of the consumers1
expenditure
variable
in equation
(iv).
with
the
new registrations
actual
Figure
3 compares
4
and Figure
(iii)
from equation
estimates
predicted
The
(iv).
predicted
from
equation
those
substitutes
of the lagged .
contribution
to the regression
negligible
dependent
variable
is obvious
from visual
inspection
of
In the French case there appears to be no
these two graphs.
justification
for inclusion
of the lagged variable.
l
The test of forecasting
reliability
was therefore
based only
on the equation which did not include registrations
for the
with
the same specification
as
previous
year
(equations
from 1965-79 to 1969-83).
(iii)
covering
15 year periods
Results of the test of forecasting
capability
are shown in
It is clear that in this French case, the model
Table 2.
failed
to provide
a warning of the sharp decrease
in the
market which occurred in 1984 and three years earlier
use of
the model would have indicated
an expansion of the market by
over 5 per cent, whereas a decrease of nearly three per cent
the model provided
remarkably
In contrast,
took
place.
close forecasts
for 1980, 1982 and 1983, still
with the
assumption
that the forecaster
was able to obtain
accurate
predictions
of the independent variables.
The French evidence reinforces
the conclusion
derived
from
the testing
of the model with United Kingdom data, that the
sufficiently
robust
to justify
further
model had proved
research into the validity
of its underlying
assumptions.
Table
2
Test of stock
France
deoreciation
model in forecastino
-Number of new reaistrations
Year
Actual
1980
1981
1982
1983
1984
1873
1835
2056
2018
1758
1 year
Forecast
earlier
1879
1971
2093
2000
1935
(000's)
made
2 years
earlier
1969
2121
2007
1930
-
-' j'
.
10
C.
*
Germanv and Belsium
German data for the years 1963-83 were used to test 'four
of the linear
stock depreciation
different
specifications
The best results
were obtained with n = 5, implying
model.
The permutations
were
a 20 per cent rate of depreciation.
(1+4A )PDI
income variable,
choice
of
based upon -the
or
(1+4h )CE (consumers'
income)
disposable
(personal
and whether
the lagged dependent, variable
expenditure),
should be included
on the right-hand
side of the equation.
Without the lagged dependent variable,
the better
result
was
obtained
using consumers' expenditure
but Ii2 was only 0.72
suggested
some autoand the Durbin
Watson coefficient
coefficient
on
The regression
correlation
of residuals.
even though
this
signed,
relative
price
was incorrectly
variable
was not significantly
correlated
with consumers'
expenditure
(the other independent variable).
Inclusion
improved
(VI
St
from ,the previous
of new registrations
the autocorrelation:R2 and appeared to eliminate
=
Coefficient
Variable
-383.71
+0.618
-t-1108.75
+0.461qtwl
Constant
(1+4A)CEt
(l+wpt
R2
where CE = ..
P
=
=
0.783
year
Std.error
of coefficient
697.65
0.299
680.21
0.207
DW = 1.90
in billions
consumers' expenditure
Deutschemarks at 1970 prices
index of prices of new cars/index
prices
- 1970 = 100
of
of consumer
The matrix
of correlation
coefficients
for equation
(v) is
from which it may be noted that the
shown in the Appendix,
is due to high
expenditure
of consumers'
insignificance
This
collinearity
between this variable
and qtml(r = 0.85).
equation
was not considered
satisfactory
enough to justify
. -
11
testing
as a forecasting
device.
been used in subsequent analysis
the stock depreciation
model.
s
However, German data have
of the basic assumptions
of
no data were found for personal
In the case .of Belgium,
disposable
income at constant prices and two equations
were
both using (1+3A)CE as the income variable
and one
tested,
for the previous
year as an
including
new registrations
Data covered the period 1966-82, and
independent
variable.
the best value
for
n was found by iteration
'to
be 4
(implying
a 25 per cent average rate of depreciation).
Without the lagged dependent,variable
the equation
produced
gave strong
an Z2 of 0.79 but the Durbin Watson coefficient
new
Inclusion
of
autocorrelation.
indications
of
year
as an additional
the
previous
registrations
in
independent
variable
improved both the overall
coefficient
of determination
and also the distribution
of residuals:Coefficient
(vi)
qt
=
y$
=
37.82
f0.210
-52.28
+0.447qt,1
0.830
P
The matrix
Appendix.
=
Variable
-
78.79
0.102
67.90
0.215
Constant
(1+3 )CEt
(1+3 Pt
DW = 1.45
consumers' expenditure
at 1970 prices
where CE =
Std.error
of Coefficient
in billions
of francs
index of prices of cars and motorcycles/index
of
consumer prices
- 1970 = 100
._
of
correlation
coefficients
is
shown
in
the
There was high collinearity
(1: = 0.89) between (1+3')CE and
in the previous
year, which reduced the
new registrations
regression
coefficient
in each case by half
compared with
those obtained when the collinear
variable
was excluded.
12
did not appear to
As in the German case, the Belgian results
the
of
justify
robust
to
any tests
be sufficiently
some
However,
of the model.
capability
forecasting
analysis
of Belgian depreciation
has been included
in the
of
the
stock
overall
assessment of the basic assumptions
depreciation
model.
3.
Tests
of the Assumntions
model outlined
.in Section
1 and
The stock
depreciation
tested in Section 2 assumes that the rate of depreciation
is
It also
constant
for all cars irrespective
of age or size.
Thus,
assumes that this rate remains constant
over time.
for all ages of car i years
l
pi
Pi-1
=
(n - 1)
n
(P = price
of one model of car)
/
where n remains constant
over the periods
and any extrapolations
regression
equations
covered by the
for forecasting.
Mogridge
(3,Appendix
3) has analysed
in some detail
the
prices
of cars of different
ages at given moments in time,
Guide Services
Ltd
(unpublished
using
data from Glass's
London
Greater
retrospectively
to
information
provided
the Motor Transactions
Survey of 1970-l
and the
Council),
His
1965-6
and 1972-3.
Surveys
of
Travel
National
conclusions
included the following:larger cars depreciate
more quickly
than small
cars and this divergence
widened from 1973 to
1981;
depreciation
period;
rates
increased
over the
1957-81
"as a first
approximationf,
median stock values
show evidence of a constant rate of depreciation
after the first
year.
13
l
Calculation
of first-year
depreciation
is complicated
by
lack of information
about actual prices paid for individual
Such
net of discount.
models of new cars, that is prices
discounts
are. likely
to vary according
to the balance
of
The subject
of discounting
was dicussed
supply and demand.
It appears
within
the industry
in the countries
concerned.
that discounting
has become a major factor
in the United
Kingdom market mainly
since
1979, though some effective
discounts
through
trade-in
allowances,
favourable
credit
terms or rebates
for cash purchases,
as well
as special
have existed
for many
for multiple
buyers,
arrangements
In France and Germany discounts
have been less
years.
partly
because of the smaller
proportions
of
substantial,
cars sold to business buyers, although there are suggestions
that the practice
of discounting
has increased
in the past
The only way to find out about actual
prices
few years.
paid for cars would be a survey of purchasers,
as the
is surrounded
by commercial
security
because of
subject
Such a
in this
particular
area.
intensive
competition
su?xey, which would require
a large stratified
sample, lies
. beyond the resources
available
for this study.
For each of the three countries,
United Kingdom, France and
of car
West Germany, it was decided to confine the analysis
prices to cars aged from 2 to 9 years.
Nine years was taken
as the maximum because model changes make it very difficult
than this
with any equivalent
new
to compare cars older
vehicle.
The simple stock depreciation
model used in Section 2 and in
previous
studies
quoted in Section
1 assumes that
in the
following
equation the coefficients
bl and b2 would be zero:
(vii)
b. + b,i + b2p
(1 - d)i =
where
d
=
rate
of depreciation
i
=
age of car in years
in any single
(integer)
year
?Y
-
.
o%~tti,
Y'Aif~h-i
.'-\
‘:?>\
.,
->J-‘
, -?c
+
9\ i,/ 2,.-2.'.v:.
.:?-i.A._
14
P
=
The simple
A.
\
price of corresponding
the size and ,lgualityl')
model also
The United
assumes that
new car
(a proxy
b. is constant
for
over time.
Kinqdom
In the case of the United Kingdom, this equation was tested
Data were .taken
from
for
the period
1972 to 1984.
Guide to New and Used Car Prices
(Blackfriars
Motorists'
Press) for the months of March, June and September of each
The dependent variable
was the ratio of the price of
year.
a second hand car to that of a vehicle
of the same model
Analysis
was confined
registered
exactly
one year earlier.*
to models available
as new at the survey date where changes
in style etc had not perceptibly
affected
prices at the time
In most cases this confined
data to
of their
introduction.
about seven years, though there were'some runs to a maximum
value of nine years.
Equation
(vii)
was then tested
with
i
x
age in years
P
=
list
price of car new in terms of 1983
general prices
(in f thousands)
(10d)i
=
ratio of price of car of age i to that
same model of age i - 1.
Observations
were pooled
results
of the computation
from 2 to 9
of
for
each calendar
year.
are summarised in Table 3.
*Because of irregular
observations
it was necessary to
adjust some data covering
slightly
longer or shorter
Care was taken to avoid distortion
by registration
periods.
Thus a car aged five years bears the
letter
changes.
registration
letter
of a car first
registered
exactly
five
years earlier.
The
Il.5
Table
Test
3
Coefficients
(std
error
in
0 :9066
_ -0;0094
(vii)
parenthesis)
b2
bl
bO
1972*
of equation
with
UK data
Numbers
Models
of
Observ
ations
-
162
0.37
(0.0011)
-0.0054
(0.0009)
13
(0.0080)
1973*
.0.8330
(0.1071)
-0.0014
(0.0153)
0.0068
(0.0108)
23
278
0.00
1974
0.8824
(0.0050)
-0.0069
(0.0008)
-0.0032
(0.0005)
20
317
0.26
1975
0.8519
(0.0078)
-0.0049
(0.0012)
-0.0033
(0.0009)
20
341
0.08
1976
0.8596
(0.0076)
-0.0060
(0.0013)
-0.0040
(0.0008)
16
291
0.15
1977
0.8664
(0.0076)
-0.0042
(0.0012)
-0.0045
(0.0006)
13
239
0.21
1978.
0,8575
(0.0097)
-0.0022
(0.0012)
-0.0057
(0.0010)
13
241
0.12
1979
.>0.8596
(0.0096)
-0.0049
(0.0010)
-0.0024
(0.0016)
16
314
0.07
1980
0.8772
(0.0101)
-0.0081
(0.0013)
-0.0051
(0.0014)
14
285
0.15
1981
TO.8537
(0.0085)
-0.0092
(0.0012)
-0.0013
(0.0012)
12
229
0.21
1982
0.8803
(0.0077)
-0.0165
(0.0012)
-0.0031
(0.0006)
10
206
0.52
1983
0.8701
(0.0011)
-0.0150
(0.0017)
-0.ogo2
(0.0013)
11
211
0.26
-0.0212
(0.0018)
-0.0001
(0.0009)
10
200
0.42
l
.
0.8760
(0.0114)
1984
*
Based
on
data
for
March
and
June
only.
_
16
(a)
Depreciation
constant
irrespective
of ace?
If the annual depreciation
rate did not vary with age then
Table 3 shows that the coefficient
bl would tend to zero.
was negative
throughout
and was significantly
different
from
The results
also
zero for all years except 1973 and 1978.
suggest that the impact of age on the depreciation
of cars _
cars
Older
over
the years.
increase
has tended
to
depreciate
at a faster proportionate
rate than cars of more
recent registration
and the difference
appears to have been
widening.
l
in as much as a combination
of
The year 1973 was unusual,
and
import
the
UK industry
difficulties
in
supply
restrictions
meant that some demand for new cars was not
Cases were reported
in that
year
immediately
satisfied.
of almost-new
second-hand
cars exceeded the
where prices
It is not surprising
list
prices
quoted by manufacturers.
of some older
cars received
a positive
that
the prices
boost.
The 1978 exception
is harder to interpret.
The distorting
effect
of the assumption that depreciation
is
constant
with respect
to age may be gauged from Figure 5.
This is based on calculations
relating
to a car worth f5,OOO
The "actualV1 depreciated
vaiues derived
new at 1983 prices.
are compared with
from the regression
results
for 1984
those which would have applied
with a constant
rate
(the
geometric
mean over the eight year period)
of 24.3 per cent
(depreciated
value = 0.757'-l
for ages i from 2 to 9).
It
should be noted that the acceleration
of depreciation
with
in the
age was greatest
in 1984, of any year included
analysis
and therefore
the distortion
would also be most
pronounced in that year.
.
.
17
(b)
Depreciation
price?
constant
irresoective
of orisinal
In the analysis
price was used rather than engine
guide to the size and quality
of the car.
model
size,
as a
the value retention
factor
(lThe coefficient
b2 relating
d) -to car price
is negative
in every year except
1973,
indicating
that more expensive cars depreciate
more quickly.
The coefficient
was significantly
different
from zero in
The
results
‘
for
1983
and 1984
eight
of the 13 years.
strongly
suggest that in those years the original
price of
It is
the car had a negligible
effect
on depreciation.
possible
that the market had.changed in the last two years.
(c)
Denreciation
constant
over time?
to
Since bl and b2 are not equal to zero, it is not possible
assess changes in depreciation
rates simply by the evolution
price
Table 4 shows for an l'average" car with list
of b,.
at 1983 prices of f5,OOO the ratios
of depreciated
values of
cars aged six and nine years to those of corresponding
These ratios
are derived
from the
models aged one year.
In addition
a notional
la-year
equations
in Table 3.
based
on an
shown,
ratio
is
depreciation
cumulative
extrapolation
of the regression
results
for years 2 - 9
the
geometric
mean rates of
Finally,
reported
in Table 3.
depreciation
over the years 1 - 9 and 1 - 12 are shown.
.
18
Table
Depreciation
4
Price
II
l
implied
bv reqression
of car aqed i years
II
II II
1 year
i = 12*
results
Annual average
depreciationf%)
1 to 9
1to
12*
Year
i-6
i=9
1972
0.423
0.220
0.103
17.2
18.7
1973
0.474
0.297
0.184
14.1
14.3
1974
0.415
0.222
0.109
17.1
18.2
1975
0.361
0.182
0.087
19.2
20.0
1976
0.361
0.179
0.083
19.3
20.2
1977
0.387
0.206
0.105
17.9
18.5
1978
0.371
0.198
0.103
18.3
18.7
1979
0.389
0.206
0.103
17.9
18.7
1980
0.369
0.180
0.080
19.3
20.5
1981
0.349
0.162
0.067
20.3
21.8
1982
0.325
0.128
0.041
22.7
25.2
1983
0.346
0.145
0.051
21.4
23.7
1984
0.308
24.3
27.5
0.029
0.108
* notional
(see preceding
paragraph)
was an
1973 which
except
in
shows that,
The table
indicated
by
rates (as
exceptional
year I depreciation
stable
during
the
were fairly
second-hand
car prices)
Between the ages of 1 and 9 years the mean rate of
1970's.
depreciation
varied within
the range 17.1 to 19.3 per cent.
Since 1981 depreciation
appears to have accelerated
and the
effect
of age is more pronounced.
_
19
(d)
How serious is the distortion
invalid
assumptions?
resultinq
from the
model assumes that in equation
(vii),
The stock depreciation
= b. + bli + b2p, the coefficients
bl and b2 are
(1 - d)i
The results
reported
over time.
zero and b, is constant
above show that all three assumptions are invalid.
Why then
does the
use of the model to explain
new
registrations
over the period 1964-84 provide the close fit
is the distortion
from
reported
in Section 2? How serious
adoption of the oversimplyfying
assumptions?
l
be important
The significantly
negative
value
of b2 would
of the car stock by size
(price)
only if the composition
changed over time and the coefficient
was so large that the
Analysis
of the car
average depreciation
rate was affected.
that
the
proportional
shows
size*
engine
stock
by
distribution
has changed insufficiently
for the small values
Through.out this
part of the
of b2 to have much effect.
validation
exercise
the value of p was assumed to be 5.0.**
are-the
changing values of
possible
significance
In order to assess the distortion
it was decided
b. and bl.
to compare the values
of depreciated
stock
implied
by
using a 25% annual depreciation
rate
equations
(i) and (ii),
for all
cars, with those derived by use of the coefficients
for each year shown in Table 3.
Of greater
vans from the Department
*Using data for cars and light
Transport
and predecessors,
published
in Transport
Statistics,
HMSO.
**f5,000
being an approximate
average new
- it should be re-emphasised
that the size
coefficient
means that the calculation
is
calculation
of
this figure
and a laborious
was not necessary.
while feasible,
price,
of
car price in 1983
of the b
not sensi z ive to
actual average
f
_
20
the
in
must
be considered
depreciation
vear
First
the
using
figure
stock
calculation
of
a depreciated
which describe
depreciation
only during
regression
results,
In 1972
the second to the ninth year of the lives of cars.
two years covered by the analysis
of
and 1973, the first
the new car market was very buoyant
and
car-price
data,
discounting
was probably
less extensive
than it has been at
In these two ,years
the observed
any subsequent
time.
average actual
ratio
of the price of one-year
old cars to
the current
price
of the same model brand new was almost
exactly
the square of the ratio derived from the equation:-
(1-d)
= b. + bl(times
1.0)
+ b2(times
5.0)
= 0.865 whereas the average
in 1972 one would derive
(l-d)
actual ratio
of the price of a one-year old car to that of a
new car was 0.750:
in 1973 the derived figure would be 0.872 whereas the actual
average ratio was 0.761.
of applying
(1-d)2
This principle,
throughout
was used
depreciation
depreciated
stock values.
to
. .
estimate
first-year
calculation
of
the
The other assumption necessary for this calculation
was that
annual depreciation
to age of
the,coefficient
bl ( relating
car) could be extrapolated
beyond the age of 9 for ages loThis assumption
is less dangerous than it may appear,
15.
because differences
in incremental
depreciation
have little
on the value of
effect
(in terms of new car equivalents)
lost over 85 per
cars which after
nine years have already
cent of their
initial
value.
1973 to 1981 the calculated
depreciation
For the years
coefficients
were applied to the distribution
by age of cars
The published
statistics
and light
vans in Great Britain.
-
:
’
--.. i._
21
of years up to age 15 and it
was
show ages by pairs
necessary to interpolate
for individual
years, a procedure
which was aided by the use in the published
tables
of
For
1982
to
1984
data
different
pairs in consecutive years.
were obtained'from
the Department of T ransport for vehicles
years up to age 15. In
of car-body type, showing individual
an average
the case of those vehicles
aged over 15 years,
stock in
age of 17 was assumed - the estimate of depreciated
new car equivalents
is not sensitive
to this assumption.
\
stock values in Table 4
The definition
of the depreciated
may be summarised algebraically
(each variable
refers to an
individual
year).
mi
=
number of cars of age i years with
licence
current
bj
=
coefficients
individual
(vii)
(l-d)i
=
b. + bli
co
=
1.0
Cl
=
PO + bl + 5b2) 2 first
Ci
=
Ci-1
2
=
number.of
while
given
=
that
by
s2
for
for
i = 2 to 17
year depreciation
cumulative depreciation
i= 2 to 17
(1-d) i
for
cars aged over 15 years.
of stock
"derived
from regressionll
15
c Cimi + Cl72
i=O
resulting
=
+ 5b2
from equation
no depreciation
on current
year's registrations
In Table 4 the estimate
is given bySl
derived
year
from "25% constant
15
c 0.75hlli
i=O
+ 0.7517Z
depreciationI'
is
-
'
22
Comparative
TABLE 5
estimates
Derived from
equation
(vii)
(S1)
Cars and light
N
1973
1974
1975
1976
1977
1978
1979
1980
1981
Vehicles
1982
1983
1984
25% constant
depreciation
(S2)
goods vehicles
(million
figures
5.95
5.80
of car body type
6.36
6.34
(million
5.46
5.71
'5.91
S2 as %
of1
equivalents)
91
89
published)
4.92
5.22
5.28
5.31
stock
77
82
4.78
4.66
.4.56
5.46
6.01
5.91
new car
4.93
6.44
5.84
5.13
5.11
(no car stock
of depreciated
88
87
89
92
new car equivalents)
92
90
93
In 1973 the prices
of second hand cars appear to have been
boosted by the delayed deliveries
of new models, and in 1974
the new car market was disrupted
by the rise in the price of
oil and its effects
both on the economics of car ownership
It is clear
from Table 5
and on the economy as a whole.
that,
except in those two years, the estimate of depreciated
stock derived
from a constant
25% depreciation
rate is very
close throughout
to 0.90 of that derived from the regression
results.
Closer correspondence
between the two estimates
was obtained
when the depreciation
rate was reduced to 22 per cent (n =
which accords with the iterative
regression
results
4.55),
of Deaton and Muellbauer
(1) up to 1975. Use of this figure
(i) and (ii)
for 1964 - 84 reduced E2 slightly
in equations
The
and the Durbin-Watson
coefficients
more substantially.
23
results
do however
demonstrate
that
the
assumption
of
constant depreciation
at somewhere within the range 20 to 25
of depreciated
stock not far
per cent produces an estimate
removed from that derived from empirical
analysis.
Table
Use of 22 per cent depreciation
6
22% constant
depreciation
(S3)
Derived from
equation
(vii)(
Sl)
Cars and light
N
1973
1974
1975
1976
1977
1978
1979
1980
1981
goods vehicles
(million
n&
6.44
5.84
5.13
5.11
5.40
5.28
5.17
5.09
5.46
6.01
5.95
5.80
5.43
5.73
5.83
5.86
Vehicles
equivalents)
of car body type
5.91
6.36
6.34
1982
' 1983
1984
car
6.01
6.26
6.49
Tests of assumptions
in overseas countries
B.
Analysis
for France, Germany and Belgium has been confined
to
to prices at one point in time but has also been applied
(1 - d)i
= b. + bli
+ b2p
where i = age in years
P = price of a new car
France
Data were taken
prices
and from Automarque
obtained
years
from 1lArqus
with
for
25 models,
97 price
(13 June 1985)
(June
extending
1985)
in
and age observations
for
for
second-hand
new.
some cases
in total.
Runs were
to
nine
24
For i = 2 to 9
the results
were
as follows:-
Standard
Coefficient
error
Comment
b0
0.855
0.014
Constant
bl
-0.0205
0.0026
Age (2 to 9)
0.00009
New price in
000 Francs
0.00034
b2
-
z2 = 0.47
l
The main difference
from the United Kingdom is the positive
that in France more expensive
cars
value of b2 indicating
The coefficient
with time is
lose their value less quickly.
shows cumulative
6, which
and Figure
similar
fairly
depreciation
in France of a car with a new price of 60,000
is remarkably
similar
to Figure
5 for the United
Francs,
of .coefficients
gives
a
Kingdom, because the combination
very similar
equation for the depreciation
rate annually
to
that for a f5,OOO car in the UK in 1984.
France:
(1 - d)i
= 0.875 - 0.0205i
UK:
(1 - d)j, = 0.876 - 0.0212i
is no information
about discounting
in France - the
price- of a one-year old car was on average only 68% of that
slightly
less
than
the
of the
corresponding
new car,
corresponding
UK average of 70 per cent.
If,
as we were
told,
discounting
is less prevalent
in France, this suggests
that the loss of value in France during the first
,year is
relatively
greater.
There
Germany
Runs
Data were taken from DAT-Marktspiegel
for March 1985.
in some cases to nine
were obtained
for 23 models, extending
the total
number of observations
was 97.
years;
were as follows:For i = 2 to 9 the results
.
-
25
Standard
Coefficient
b0
0.900
0.013.
Constant
bl
-0.0093
0.0019
Age (2 - 9)
b2
-0.00026
0.00025
New price in
000 marks
(not significant)
R2
\
Comment
error
=
0.20
suggesting
that new price
This time b2 is not signif icant,
has no effect
on depreciation
rates - consistent
with the
1984 evidence
for the United Kingdom but contrasting
with
In the German case (see Figure 7, which applies
to
France.
lower
rate
of
new price
of DM20,OOO) the
a car with
the
depreciation
with
acceleration
of
age means that
assumption
of constant
depreciation
over the range 2 to 9
years is less invalid
(it should be noted that the average
depreciation
rate over this age range is only 15 per cent
There is no information
about discounting
in
per annum).
Germany - the average price of a one-year old car was only
67 per cent of that
listed
for a corresponding
new car.
This proportion
is higher than in the UK and France, despite
The
the subsequently
much lower rate
of depreciation.
combination
of the higher
first-year
depreciation
and the
much lower subsequent figures
may explain why German data do
not fit satisfactorily
the depreciated
stock model, with its
is constant
with
respect
to
assumption
that
depreciation
age.
Belsium
Data were taken
from a guide published
by Autokrant
in
Antwerp
showing
the prices
net of tax in May 1985 of
individual
models of cars first
registered
in the years 1977
are based on vehicles
of average
The figures
to 1983.
mileage
(between 10 and 20 thousand kilometres
a year).
The reason for using
figures
net of tax is that
in Belgium
there
is a registration
tax payable
on the purchase
of
. -
.
;>.-
I
.
:
”
:
.
_
26
.
second
nominal
general
second
easily
vehicles.
%
on the basis
of
which is calculated
hand cars,
dehreciation
from the original
price
(corrected
for
The net prices for consecutive
years of
inflation).
they cannot
are directly
comparable;
hand cars
be compared with the prices of the corresponding
new
Runs of data for six or seven years were obtained
for 56
car models with ages from 3 to 8 years and 334
individual
the
consistency
of
To test
total.
observations
in
depreciation
of car prices of Belgium with that observed in
we regressed
price ratios
against the age
other countries,
The
of the car and also the price of a two-year old model.
results
were as follows:Standard
Coefficient
error
0.113 -
Comment
Constant
b0
0.828
bl
-0.0106
* 0.0014
Ages 3 to 8
b2
+0.0045
0.0016
Price of 2year old car
in 100,000
Francs
3
=
0.16
As in France the coefficient
b2 is positive
- more expensive
depreciation
Although
slowly.
more
depreciate
cars
accelerates
with age this effect
is less pronounced than in
any .of the other.three
countries
(see Figure 8 which refers
to a car with
a two-year
old price
in 1985 of 400,000
year
No information
is available
about first
Francs).
and it is therefore
not possible
to analyse
depreciation
explain the relatively
poor fit of
whether this might partly
the model in the Belgian case.
4.
Some Conclusions
In the case of the United Kingdom the detailed
depreciation
of cars as revealed by secondhand
analysis
of
prices
shows
*
27
:
that two basic assumptions of the stock depreciation
model
There is significant
statistical
evidence that
are invalid.
the rate at which the value of a car depreciates
increases
with its age and also that depreciation
rates have tended to
Germany and
Evidence from France,
increase
over time.
during
1985,
which relates
only to car 'prices
Belgium,
confirms that depreciation
increases with age but the nature
of this relationship
differs
between the countries.
t
The relationship
between depreciation
and the quality
of th
of a cxband new
by the purchase price
car (as indicated
vehicle)
is not consistent
between the four countries.
In
?.I
the case of the United Kingdom more expensive
cars have
tended to lose their
value more quickly
although
this
feature has lost statistical
significance
in 1983 and 1984.
In France more expensive
cars appear to depreciate
more
is also suggested by the evidence
from
slowly
and this
in Germany the price .of the car does not seem to
Belgium;
affect
the rate of depreciation.
two previous
results
summarised in the
The regression
paragraphs do not include depreciation
during the first
year
Because of lack of information
about
life.
of a vehicle's
it is not possible
to
estimate
the scale of discounting,
evidence from
first-year
depreciation
with any accuracy;
in the UK is believed
to
1972 and 1973, when discounting
!
first-year
that
suggests
minor,
have been relatively
double that which occurs in
depreciation
is approximately
Estimates of depreciated
car stock based
the second year.
on a constant
depreciation
rate were found to be fairly
close to those based on the regression
results
for the
of double
with
the added assumption
United
Kingdom,
This may explain
why the
depreciation
in the first
year.
The
stock depreciation
model gives a reasonably
good fit.
correspondence
of the two values may be somewhat fortuitous,
reflecting
the distribution
of the car stock by age over
In Germany, where the assumption
this particular
period.
28
that depreciation
does not change with the age of the car is
fit,
satisfactory
the model gives
a less
more valid,
presumably because it is less able to accommodate variations
problem
of
the
time
and also
over
depreciation
in
disproportionate
depreciation
in the first
year.
l
The stock depreciation
model l,workslV for the United Kingdom
-7
simple regression
and France, in the sense that a relatively
model based upon it explains
most of the variation
in car
It also
registrations
over
a period of about twenty years.
i
._
l~works~~ as a forecasting
method, assuming that forecasters
had been able to predict
changes in consumers'
expenditure
However, the conclusions
and in the relative
price of cars.
of the analysis
of car prices
suggest that further
work is
.
necessary to explain why the model does "work*' so well even
This
though its
fundamental
assumptions
may be invalid.
further
work would need to focus on first-year
depreciation
and also on the mechanism whereby depreciation
in stock is .
Direct
surveys
translated
into purchases of new vehicles.
for
this
purpose.
would be required
of car purchasers
there is a case. for further
regular
testing
of
Meanwhile,
the model and possibly
its tentative
use in forecasting.
29
CORRELATIONMATRICES
APPENDIX 1
UNITED KINGDOM
New Car
Ress (t)
Relative
Price
+
New Car
Registrations
(+I
0.139
$
Consumers'
Expenditure
Consumers*
Expenditure
Relative
Price (+)
0.964*
0.277
(t-l)
0.773*
-0.073
+
0.265
0.733*
FRANCE
*
Relative
Price
Consumers'
Expenditure
+
New Car
Registrations
0.970*
0.381
(t-l)
O-899*
0.158
+
0.272
.
0.919*
WEST GERMANY
Relative
Price
Consumers'
Expenditure
+
New Car
Registrations
(t-l)
0.833*
0.041
0.860*
0.120
0.843*
BELGIUM
Relative
Price
Consumers'
Expenditure
New Car
Registrations
+ This variable
* Significantly
-0.131
+
+
(t-l)
0.892*
0.012
0.904*
-0.114
specified
different
0.885*
throughout
in the (1+3h) format.
from zero at the 5 per cent level
(or less)
REFERENCES
(1)
J. Economics of Consumer
Deaton, A. & Muellbauer,
Cambridge University
Press, Cambridge, 1980
Behaviour.
(2)
Dicks-Mireaux,
L.A. t OlHerlihy,
C.St.J.
Prospects
National
Institute.Economic
the British
Car Industry.
Review, 1961
(3)
Mogridge,
(4)
Nerlove,
M.
Econometrica,
(5)
OlHerlihy,
C.St.J.
Applied Statistics,
(6)
Consumer Demand for Cars in the USA,
Smith, R.P.
Cambridge University
Press, 1975
(7)
Stone, R. & Rowe, D.A. The Market
1957.
Goods, Econometrica,
M.J.H
The Car Market,
Pion Ltd.,
London 1983
l
The Market
1960.
Demand for
Demand for
1965
Durable
for
Goods.
Cars in Britain.
Demand for
Durable
FIGURE 1
U.K.:
equation
TEST
OF MODEL
W ITH
N = 4:
‘1964-84
(i)
-6-l
-!J
L”
-4
u-l
-z-i
if
L
x Actual
dL
u
(with
+ Estimated
0
1964 I
I
I
1966
8
I
1968
continuous
<with
b
I
1970
broken
i972’
Years
line>
line>
is74 I
i.976 i
/
I
1978
i
99a’
6
I
199-L‘
‘,
1924
,
FIGURE 2
U.K.:
equation
TEST
OF MODEL
W ITH N = 4:
1964-84
(ii)
L
-a
VI
G
750,
L”
L
x Actual
<with
+ Estimated
continuous
(with
broken
line>
Line>
2501
1964 I
I
I
1966
1
I
1968
I
1970
I
i 972’
Years
i974 ,
/
1976
,
I
1978’
issa’
i 932’
I
1994
FIGURE 3
FRANCE:
equation
TEST
OF MODEL
WITH
N = 4:
1965-80
(iii)
iii
ks
w
:
0
“A
4
F!
-4
VI
-t-l
ii?
L
750
iJ
dL
U
x Actual
(with
+ Estimated
continuous
(with
line>
broken
line>
500:
z
Z
250:
0
1965 I
1
1967’
/
I
1969
i
1
1971
I
4
1973
Years
i975 8
i977'
i979
I
i99f
i 993
I
.
.
FIGURE 4
FRANCE:
equation
TEST
OF MODEL
WITH
N = 4:
1965-84
(iv>
2250
I
k
U
2
(with
x Actual
+ Estimated
continuous
(with
line>
broken
line)
500:
Z
250:
0’
1965 ,
I
#
1967
I
,
1969
I
i
1971
I
,
1973
Years
I
t
1975
i977 /
i979’
1931’
I
i
993
I
_
.
i.
,.,.
.,.
.
I
I,
1.
.,..
.
.
t ,
F IG U R E
5
U K C U M U L A T IV E
D E P R E C IA T IO N
IN
1984
1 .0 0
-
L
d
%
0 .9 0 1
G !l
0.80:
E
,I-4
k 0.70:
l
-
E
t
-zA
4
d
-l-l
z
k
G?
0.60:
0.50:
0.40:
0.30:
:
*l--4
4
26
3
5
0
I
I
/
I
I
/
L
1
2
3
4
5
6
7
8
9
Years
sin c e
F it-st
r e q istra tio n
FIGURE6
FRANCE:
CUMULATIVE
DEPRECIATION
0.90
@I
E!i
-P-i
k
0.80:
* :
0.70:
:
t
k
-I+
-4
d
-l-l
t
k
lsi
0.60:
0.50:
0.40:
0.30:
>”
-A
-d
d
d
2
3
0.20:
Years
since
First
registration
IN
1985
FIGURE7
GERMANY‘:
L
6
1 .00,
CUMULATIVE
DEPRECIATION
IN
1985
1
%
f
0.90:
:
QJ
@
U
.A
k
0.80:
* I
0.70:
6
L
L
0.60s
5
2
0.50:
.t-l
U
f
0.40.
:
:
-0
.t-i
t,
d
J
3
0.20:
5
0
0.10:
/
:
0.00;.
0
11
2I
Years
I
5
I
4
3I
sii?ce
First
1
6
registration
i
7
I
8
r
c
FIGURE8
BELGIUM:CUMULATIVE
DEPREClATlON
IN
1985
X10”
IIO,
cx
i.2
0
-I
0
cc
zi
>
l
A
9:
7:
x
.O
oz
IL
6:
0”
C
+
a
C
5:
El
E
4..
E
W
>
l--l
+
a
2
x
3
0
31
2:
1:
0,
0
I
1
t
2
YEARS
I
3
SINCE
,
4
FIRST
I
5
REGlSTRATlON
L
6
I
7
il