ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
IMPROVING THE INFORMATIONAL CONTENT OF THE SIMPLE
CAPITAL BUDGETING MODEL
by
Peter Brownell
WP 1173-80
November 1980
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
IMPROVING THE INFORMATIONAL CONTENT OF THE SIMPLE
CAPITAL BUDGETING MODEL
by
Peter Brownell
WP 1173-80
November 1980
Improving the Informational Content of the Simple
Capital Budgeting Model
The
purpose
improvement
in
of
the
this
paper
is
information
random variable rather than as
offer
content
budgeting model via the treatment
a
to
in
a
a
of
the model
suggestion
the
simple
of cash
an
capital
inflows as
single-point estimate.
I^OT^
for
-1
One
budgeting model
capital
first
tool
is,
accounting
year
Unlike
impact
simple
approach
model,
where
of
discount rate
for
volatility
the
of
namely
a
employed
in
approach
order
I
some
and
analysis
This
inappropriate
of
sales
will
to
introduce
or,
purpose
and
relevant approach
to
the
a
the
reveal
the
volume,
the
analysis
changes
to
Why
and
volatile
more
generally,
in
the
that
the
element
of
the
cash
flow,
it
is
is
mystery.
vagaries
of
the
traditional
simple example which shall also serve
detailing what
the
for
a
value
period
one
to
sales
of
flow pattern.
important
illustrate
geared
is
level
present
net
volume
simple,
the
budgeting-investment
given cash
most
in
the
in
traditionally overlooked remains
In
year
decision
inclusion
analysis
the
capital
relates changes
analysis,
second
long-term
a
the
to
changes
multi-period
analogue
due
the
profits
on
as
most
in
simple
the model.
in
break-even
courses
partially
feel,
variables
introduced
inadequacies of the
failure to deal with conditions of uncertainty.
its
is
I
major shortcomings or
the
of
I
consider
a
far more
decision analysis.
significant
-2-
X
Proposal
is
produce
to
Co. Ltd.
.
and
company is faced with
The
machines,
prod uc t ion
A
:
and
a
to
B,
sell
"gizmos"
at
choice between
acquire
for
purposes
fixed costs (all cash)
3y
Estimated Salvage Value
sold annually.
its capital
follows
:
surveys
10^
1Q6
90^
30^
$1,000
$10,000
Expected Life
reserach
$7,500
$20,000
Contribution margin per unit
3yr
r s
suggest
25,000
a
s
$0
$0
The company employs
units
will
be
10% discount rate
project evaluation.
traditional
The
of
B
A
prod uct
in
different
2
Variable cost to make and sell
Market
each.
-
Cost of Machine
Annual
$1
investment
analysis
proceed
would
as
-
A
Expected Revenue (annual)
V z,r
iab le
Cost
s
B
$25,000
$25,000
2,500
17,5 00
500
Cjntribution Margin
22,500
7,
Annual Fixed Costs
10, 000
1
12, 500
6,500
Net
Annual Cash Flow
Annuity factor
,000
2.4868
2
Present Value of Cash Flow
31,085
16,164
Purchase Price
20,000
7,500
X
Net Present Value
at
10 ?c
for
3
years
$11,085
$
.4868
8,664
3-
On
10
?c
basis
the
machine
,
reveals
that
either
at
preferred.
In
fact
by
of
any
the
use
ends
there
machine
and
levels?
sales
85c
discount
choice
the
rate
made.
is
would
choice.
estimate
sales
introducing
is
particularly
is
shaky
(of
situation so too would be the estimate of
un it
)
relevent
seems vastly more
impact
value
present
net
on
in
rates
discount
what
choice
onto
analysis
The
0*.
various
of
expected
if
market
the
course
event
the
In
the
such
in
a
sales price of $1 per
a
discount
holding
value
sales
at
fixed
at
at
and
10%
25,000
various discount
estimate
sales
fixed
rate
the
varies
and
units
rates.
But
surely
management
sales
level(s)
I
propose holds
rather
sales,
than
present
net
might be suggested that the
chosen
rate
would
be
evaluative
for
With this no one can
purpose are both very subjective in nature.
argue.
consider
computing
and
It
discount
varying
analysis
The
to
instance
this
cf
(N)
rather than varying discount rates.
the
revealed
.
It
the
will be
30,000?
Or
product
new
a
B
But
preferred
the
be
clearly
various
At
of
still
is
2
rate
Further analysis
A
than
less
indicated
the
What
firm
the
discount
a
machine
12"n
or
sales were only 20,000 units annually?
that
at
,
preference for machine
no
always
is
A
value
clearly the preferred choice.
is
A
present
net
of
as
concerned
with
the
impact of varying sales levels as with different discount rates.
The
sales
are
analysis
plotted
is
on
best
the
performed
abscissa
graphically and
and
net
present
in
Figure
value
on
1
the
.
•
-4-
The mathematics of the
ordinate.
be
value
performed
analysis
The
it
present
net
seen,
that
presented
is
at
Figure
reveals
1
A
level.
form
and
25,000 units per
of
exceeds that for machine
B.
information relevant
to
host of additional
a
sales
graphical
in
sales level
a
present value for machine
annum net
function of
linear
a
above
can be clearly seen
But
is
analysis are simple for, as can
the cho ice
Notice,
present
made.
Not
"break -ev en"
units
13,386
clue
Second,
while
machine
fcr
fact
20,047;
is
it
A
reveals
for machine
to
1%
be
the
in
the
fact
is
only
B
which
fact
a
net
only
be
to
revealed
was
chart
the
the
wrong choice
the
present value)
net
has
below
fall
to
preferred by
be
for
this
to
only
has
estimate
analysis
however.
(zero
to
B
sales
The
slightest
analysis
earlier
that
the
level
machine
for
traditional
the
in
sales
the
criterion.
value
high
that
annually
units
23,378
too
first,
management might be impressed with if the sales level was utterly
uncer ta
in
It
seems
proposition
that,
than
machine
Notice
B.
23,378 units the "cross-over point"
sales
were
same
and
23,378,
our
indifference
below
the
machine
A
net
cross-over
changes
by
the
analysis
two.
point
a
we
But
see
proportion
that
A
at
occurs.
present value
traditional
between
machine
sense,
some
in
for
that
a
That
riskier
far
a
level
sales
if
is,
suggest
sales
net
greater
a
of
expected
each alternative
would
for
is
is
the
state
of
levels
above
and
present
value
for
that
for
than
does
-5-
machine
machine
for
and
A
levels
sales
for
A
shall
I
refer
are
we
cross-over,
above
This
risk.
as
to
quantified quite handily by the slope of the curves
machine
For
A
marginal
a
produces
for machine
$2.24 while
present value of
sold
unit
worse
far
far
sensitivity of net present
This condition of higher
better off.
value
sales below cross-over
for
is,
machine
with
off
That
B.
be
the chart.
in
incremental
an
is
75 c
B
can
net
produced.
We
might define these as "risk coefficients" and, other things being
equal,
management
conceivably
might
between
indifferent
be
projects with different risk coefficients, thus displaying
indifferent nature.
alternative
the
more
To
typical
state
incorporate
nature
such
of
traditional
uncomplicated analysis.
part
the
on
in
from
cry
our
relevance of such
the
But
otherwise)
(or
far
a
a
management.
of
averse
risk
analysis
the
in
the
as
displaying
coefficient,
risk
aversion
risk
factor
a
management
of
smaller
the
risk
however, would be the choice of
More likely,
with
a
two
a
factor cannot be denied.
How
Machine
value,
Machine
does
A
or
B
preferred)
decision
our
preferred
is
what
shall
I
is,
however
and
also
the
on
refer
the
seems that
analysis
in
order
must
to
in
a
its
-
u
Let
expected
NPV
is
derive
a
any
present
net
to
sense
risk-averse)
How is cur decision
attach
recap.
us
preferred
"break-even"
management
(assuming
we
being
to
now?
look
basis of
preferred
risk sense (risk preferred).
It
problem
probabilistic
to
B.
(B/E
in
a
be made?
dimension
rigorous decision.
We
2
to
will
-6-
recall
sales
that
(y-sales)
machine
preferred
be
probability
the
had
distribution
such
of
sales
of
terms
in
probability
the
may
indiciation of
some
about
statistically,
or
ii,
deviation
sales
of
which
units,
well
as
variation
like
high
sales
of
know
to
expected
their
as
already know.
we
would
we
around
u
in
would like to
We
v.
the
is
standard
the
value
levels
25,000
of
introduces difficulties into
This
analysis for, without knowledge of the distributional form of
the
sales
its
greatly
the
standard
also
and
y
elicit an estimate of the
market
are
once
25,000
sales
units
to
annually,
be
sales
(a).
exact)
within the range
y
-
of
la.
purposes
our
sales
that
provides
albeit
for
asked
and
are
This
generally
we
crude,
you
an
will
recall
the
area
under
introduce
normally
opportunity to
sales manager
estimate of y=25,000
66
what
2/3?o)
estimated
We
cannot
computational
The
"within
(i.e.
2/3
provides
the
I
are
for
standard deviation.
consulted
fall?"
deviation of
(68.26?c
This
research team responsible
again
will
For
convenient,
a
etc.)
assumption
25,000.
of
allows
gamma,
deviation.
simplifying
distributed with
ease
poisson,
normal,
(i.e.
discover
or
expected
the
that
quite
be
that
Perhaps
value.
low,
amount
order
concentrated
highly
very
is
very
is
the case of marked dispersion of sales about
have
expected
their
present
net
of
outcome
an
Alternatively,
(25,000).
below
fall
to
25,000 units annually to 23,378 units in
of
B
only
that
the
of
range
about
certain
the
standard
approximately
normal
that
curve
68%
lies
,
-7
Suppose
thst
related
information
vital
sales
have
only
desirable
more
that
learn
fall
to
achieve "break-even"
is
undesirable
qualities
with
and
than
machine
less
far
failing
failure
The
to
higher
a
this
in
case
sufficiently
for
than
a
machine
management
percentage
37?o
Clearly,
more
likely
to
cannot
reach
5
is
as
in
A.
have
may
a
standard
probability
tne
so
A
increases
clear
and
to
0.16.
management
characteristics
a
new present
this
percentage
choose
%
units
value
might
inspite of the fact that it
B
chance of producing
A.
present
sales jumps to 0.37 and
machine
averse
risk
case,
the
"break-even"
displaying
a
level
not
is
conceivably elect to choose machine
offers only
such
In
achieve "cross-over"
break-even
ignore
to
net
25,000
distribution
the
inadequate
level
respect
of
to
approximately zero.
expected
estimate
sales
be
only
is
sc
probably choose machine
89 will
to
B
delighted
virtually
to
with
A
5,C0C units.
achieve
decision
displaying
certain,
say,
to
machine
C.
sales
a
that
fact
machine
doing
the meantime,
position
a
of
the
their
of
in
offering
A
contrast,
deviation of,
of
in
with probability
B
By
been
thus
is.
for
possibly
now
probability of
The
Management
order
in
Some
available.
now
is
disquieting
the
are
A,
probability
the
approximately C.2I.
to
23,378
to
machine
than
about
units.
2,000
be
to
decision
the
to
concerned
perhaps
Management,
estimated
thus
is
o
machine
this
of
distribution since u-sales exceed cross-over sales.
higher
nears
Of
B.
case
value
a
course
5
%
the
symmetric
•8-
handy
A
visual
aid
analysis
this
in
is
superimpose
to
probability distribution on the graph presented
resulting presentation appears
attention
shown,
area
probabilistically
can
"relevant
range"
model
the
the
added
presented
information
significant
Three
value.
of
and
assumptions
summarization.
First,
discount
is
rate
expositions
traditionally
However more
to
discredit
in
order
it
to
develop
can
As
was
mentioned,
more
but
the
potential
refinements
are
the
where expected
empirical
model
is
sales volume
issues.
workable,
is
assumed
appropriate
in
to
its
be
beginning
is
Secondly,
are
with
any
normally
distributional
and
a
more
perhaps
most
form,
only
each
year
present
equal
rate
subjective.
involved
Finally,
of
Textbook
statements
sales
of
chosen
the
explanation.
that
is
discount
area
that
worthy
are
highly
finance
the
assume
to
a
be
complex
information
the
These
probability
useful
necessary
accurate model
importantly,
any
in
from
is
determinable.
text-book
traditional
the
above
model.
choice
the
the
additional
one
and
establishing
literature
recent
is
exist
that
the
model
the
by
and
objectively
of
out
believe
I
necessary to assume that
is
means
suggest
distributed.
forms
it
fairly
the
of
simple
the
of
But
assumptions
added
information are necessary over
item
ease,
available
made
that
only marginally mere
is
The
3a).
budgeting analysis.
capital
simple
than
have
I
1.
only
on
volume
sales
of
-
(y
blocking
By
2.
focused
be
19,000 to 31,000 units annually
The
Figure
in
Figure
in
the
for
in
.
-9-
of
life
the
were
value
project.
the
of
assumed
in
the
service life
Equal
model
only
to
simplify
salvage
zero
and
analysis.
the
These can be incorporated without much difficulty.
informational
added
only
The
need
estimate of the standard deviation of the
suggestion for
is
an
sales distribution.
A
simple (albeit crude) means of establishing this
a
made
wa s
conclusion
In
simple
finance,
sophistication
introduced
an
is
very
incomplete
the
they
not
popularized
are
simple
worthwhile
example of
simplifying
the
should be said
it
decision models
business
of
model
the
of
a
and
inadequate.
in
for
in
decision
asssumptions of
that
the
as
in
management
one
many
good
cases.
problem
for
case of many
the
accounting
have
which,
traditional model
Moreover,
the
added
reason:
I
added
proposed approach is provided at minimal cost.
and
I
feel
however,
render
it
usefulness
-10-
Net
present
Value
$'000
-11-
Net
present
Va] ue
20
$'000
5000
10000
15000
20000
19000
M-3a
FIGURE
2
The "Relevant Range" Depicted
25000
30000
35000
31000
u+3o
Sales per year in units.
u
-l:
FOOTNOTES
I
Mathematically, the partial deriviative of each machine's
present value function produces the above results: Machine A 3N
net
:
N
d
(
.9s
1
-
f
1
Machine
N„
where
Pi
)
1
d
(
N
d
as
9d
=
.9
x
2.
4868
=
$2
•3d
=
.3
x
2.
4868
=
$0. 75
.
.
24
-
B
:
.
3s
3N
-
f
?
)
-
p
.
2
2
3s
is net present value
annuity factor
is the
for
the
10%
discount
rate
i.e.
2.4e68
s
f
p
is
is
is
sales level
annual fixed cost
purchase price
the
the
accommodation of probability distributions for the variables
See
involved in a capital budgeting model is not a novel idea.
for example D. B. Hertz ("Risk Analysis in Capital Investment",
No.
Jan -Feb, 1964, pp.
1,
42,
in Harvard Business Review Vol.
95-106) who outlined a procedure for attaching probabilistic
dimensions to no less than nine relevant variables in generating
a single probability distribution for rate of return.
"The
;
has been empirically demonstrated that, in fact, the logSee
normal distribution of sales is an accurate approximation.
A.,
H.
Holt, C. C., Modigliani, F., Muth, J. F., and Simon,
"
(PrenticePlanning Production, Inventories and Work Force"
Hall 1960) p. 283.
It
,
This idea is borrowed from Jaedicke, R. K., and Robichek, A. A.,
"Cost-Volume-Profit Analysis Under Conditions of Uncertainty",
T he Accounting Review
October, 1964 pp. 917-926.
,
for example, Rubinstein, M., "A Mean-Variance Synthesis of
Corporate Financial Theory", Journal of Finance
March 1973.
See,
,
Date Due
vc
MAR Gs
198|5
SE
U ^ 83
ty
-
BBS
SEP'3
4$
OCT
*W
2
\
1986
I
Lib-26-67
:o., inc.
DEC 8
100
CHA
(
1983
IDGE STREET
.
MASS.
no. 1 165- 80
McKersie, Robe/Chanqe and continuity
74 1433
D*BKS„
0Q1.3 23 79
HD28.M414
lllllfll
3
TOAD DOl
^2
630
HD28M414
Brownell,
740979
no. 1 174- 80
Pete/Leadership behavior,
D*BKS
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b