MANAGEMENT SCIENCE
Vol. 11, No. 10, August, 1066
Printed in U.S.A.
STOCHASTIC DECISION TREES FOR THE ANALYSIS
OF INVESTMENT DECISIONS*
RICHARD F. 1 IESPOSf ano PAUL A. STliASSMANN|
T h i s p a p e r d e s c r i b e s an i m p r o v e d m e t h o d for i n v e s t m e n t d e c i s i o n m a k i n g .
T h e m e t h o d , w h i c h is called the s t o c h a s t i c d e c i s i o n tree m e t h o d , is p a r t i c u l a r l y
a p p l i c a b l e to i n v e s t m e n t s c h a r a c t e r i z e d by high u n c e r t a i n t y and requirin g a
s e q u e n c o of r e l a t e d d e c i s i o n s to be m a d e o v e r a period of t i m e . T h e s t o c h a s t i c
d e c i s i o n tree m e t h o d builds on c o n c e p t s used in the risk a n a l y s i s m e t h o d and
t h e d e c i s i o n tree m e t h o d of a n a l y z i n g i n v e s t m e n t s . It p e r m i t s t h e u s e of
s u b j e c t i v e p r o b a b i l i t y e s t i m a t e s or empirical f r e q u e n c y d i s t r i b u t i o n s for s o m e
or all f a c t o r s a f f e c t i n g t h e d e c i s i o n . T h i s a p p l i c a t i o n makes it p r a c t i c a b l e to
e v a l u a t e all or n e a r l y all f e a s i b l e c o m b i n a t i o n s of decision s in the d e c i s i o n
tree, t a k i n g a c c o u n t of b o t h e x p e c t e d v a l u e of return and a v e r s i o n to risk, t h u s
a r r i v i n g a t a n o p t i m a l or near o p t i m a l s e t of d e c i s i o n s. S e n s i t i v i t y a n a l y s i s of
t h e m o d e l c m h i g h l i g h t f a c t o r s t h a t are critical because of high l e v e r a g e on
t h e m e a s u r e of p e r f o r m a n c e , or high u n c e r t a i n t y , or both. T h e m e t h o d c a n be
a p p l i e d r e l a t i v e l y e a s i l y to a wide v a r i e t y of i n v e s t m e n t s i t u a t i o n s , a n d is
i d e a l l y s u i t e d for c o m p u t e r s i m u l a t i o n .
Investment decisions are probably the most important and most difficult decisions that confront top management, for several reasons. First, they involve
enormous amounts of money. Investments of U. S. companies in plant and equipment alone are approaching $50 billion a year. Another $50 billion or so goes into
acquisition, development of new products, and other investment expenditures.
Second, investment decisions usually have long-lasting effects. They often
represent a "bricks and mortar" permanence. Unlike mistakes in inventory decisions, mistakes in investment decisions cannot be worked off in a short period
of time. A major investment decision often commits management to a plan of
action extending over several years, and the dollar penalty for reversing the decision can be high. Third, investments are implements of strategy. They are the
tools by which top management controls the direction of a corporation.
Finally, and perhaps most important, investment decisions are characterized
by a high degree of uncertainty. They are always based on predictions about the
future—often the distant future. And they often require judgmental estimates
about future events, such as the consumer acceptance of a new product. For all
of these reasons, investment decisions absorb large portions of the time and attention of top management.
Investment decision-making has probably benefited more from the development of analytical decision-making methods than any other management area.
In the past 10 or 15 years, increasingly sophisticated methods have become available for analyzing investment decisions. Perhaps the most widely known of these
new developments arc the analytical methods that take into account the time
* R e c e i v e d F e b r u a r y 1905.
t M o K i n s e y a n d C o m p a n y , Inc. , N e w York
J N a t i o n a l D a i r y P r o d u c t s C o r p o r a t i o n , N e w York
B-244
T H E ANALYSIS
OF
INVESTMENT
DECISIONS
JB-245
value of money. These include the net present value method, the discounted cash
flow method, and variations on these techniques. [4, 13] Complementary to these
time-oriented methods, a number of sophisticated accounting techniques have
been developed for considering the tax implications of various investment proposals and the effects of investments on cash and capital position. [2, 12, 16]
Considerable thought has been given to the proper methods for determining the
value of money to a firm, or the cost of capital. [12, 13] The concepts of replacement theory have been applied to investment decisions on machine tools, automobile fleets, and other collections of items that must be replaced from time to
time. [16]
In a somewhat different direction, techniques have been developed for the
selection of securities for portfolios. These techniques endeavor to select the best
set of investments from a number of alternatives, each having a known expected
return and a known variability. [11] In this context, the "best" selection of investments is that selection that either minimizes risk or variability for a desired level
of return, or maximizes return for a specified acceptable level of risk. (In general,
of course, it is not possible to minimize risk and maximize return simultaneously.)
The application of these techniques to corporate capital budgeting problems is
conceivable but not imminent.
In the evolution of these techniques, each advance has served to overcome
certain drawbacks or weaknesses inherent in previous techniques. However, until
recently, two troublesome aspects of investment decision making were not adequately treated, in a practical sense, by existing techniques. One of these problems was handling the uncertainty that exists in virtually all investment decisions. The other was analyzing separate but related investment decisions that
must be made at different points in time.
Two recent and promising innovations in the methodology for analyzing investment decisions now being widely discussed are directed at these two problems.
The first of these techniques is commonly known as risk analysis; [6, 8] the second
involves a concept known as decision trees. I'J, 10, 15] Each of these techniques
has strong merits and advantages. Both are beginning to be used by several
major corporations.
It is the purpose of this article to suggest and describe a new technique that
combines the advantages of both the risk analysis approach and Die decision tree
approach. The new technique has all of the power of both antecedent techniques,
but is actually simpler to use. The technique is called the stochastic decision tree
approach.
To understand the stochastic decision tree approach, it is necessary to understand the two techniques from which it was developed. A review of these two
techniques follows.
A Review of Risk Analysis
Risk analysis consists of estimating the probability distribution of each factor
affecting an investment decision, and then simulating the possible combinations
of the values for each factor to determine the range of possible outcomes and the
B-246
RICHARD F
IIESP0S AND PAUL A
STKASSMANN
FIGURE 1
TWO
ANALYSES
FACTOR
OF
AN
INVESTMENT
" B E S T GUESS" ANALYSIS
PROPOSAL
RISK A N A L Y S I S
PROBABILITY
Size of
Investment
Net Annual
Savings
L i f e of
Investment
10 y e a r s
Net
Pre ent Value
probability associated with each possible outcome If the evaluation of an investment decision is based only on a single estimate—the "best guess"—of the value
of each factor affecting the outcome, the ìesulting evaluation will be at best incomplete and possibly wiong This is tiue especially when the investment is large
and neither cleaiiy attractive nor cleaily unattiactive Risk analysis is thus ap
important advance over the conventional techniques The additional information
it provides can be a gieat aid 111 investment decision making
To illustrate the benefit of the risk analysis technique, Figuie 1 shows the results of two analyses of an investment proposal Fust, the piopo&al was analyzed
by assigning a single, "best guess" value to each factoi The second analysis used
an estimate of the probability disti ìbution associated with each factor and a simulation to determine the probability disti ìbution of the possible outcomes
The best-guess analysis indicates a net piesent value of $1,130,000, whereas
the nsk analysis shows that the most likely combination of events gives the pioject an expected net piesent value of only $252,000 The conventional technique
T H E A N A L Y S I S OF I N V E S T M E N T
JB-247
DECISIONS
FIGURE 2
U S E OF S E N S I T I V I T Y
A N A L Y S I S TO H I G H L I G H T C R I T I C A L
FACTORS
AN U N F A V O R A B L E C H A N G E
O F 10 P E R C E N T I L E S
F R O M T H E M E A N V A L U E IN T H I S F A C T O R
A n n u o l net c o s h flow
Soles level
12
17
S e l l i n g price
10
21
58
Manufacturing cost
F i x e d cast
A m o u n t of i n v e s t m e n t
L i f e of i n v e s t m e n t
^
5
12
6
12
30
tails to take into account the skewed distnbutions of the various factois, the
mteiactions between the factois, and is influenced by the subjective aspects of
best guesses Fuitheimore, the conventional analysis gives no indication that
this investment has a 48 peieent chance of losing money Knowledge of this fact
could gieatly affect the decision made 011 this pioposal, particulaily if the investor is conservative and has less risky alternatives available
The risk analysis technique can also be used for a sensitivity analysis The
purpose of a sensitivity analysis is to deteimme the influence of each factor 011 the
outcome, and thus to identify the factois most critical 111 the investment decision
because of then high levei age, high unceitainty, 01 both In a sensitivity analysis,
equally likely vanations 111 the values ot each factor aie made systematically to
deteimme then effect 011 the outcome, 01 net piesent value Figure 2 shows the
effect of individually vaiying each input factoi (seveial of which aie components
of the net cash inflow)
This analysis indicates that manufactuiing cost is a highly entical factoi, both
111 levei age and uncertainty Knowing this, management may concentiate its
effoits on 1 educing manufactuiing costs 01 at least 1 educing the unceitainty 111
these costs
Risk analysis is rapidly becoming an established technique 111 Amencan 111dustiy Seveial laige coipoiations are now using vanous foims of the technique
as a legulai pait of then investment analysis pioceduie [1, 3, 7, 17, 18] A backlog of expei lence is being built up 011 the use of the technique, and advances 111
the state of the ait aie continually being made by useis For example, methods
have been devised foi lepiesenting complex mtenelationships among factois
Improvements aie also being made 111 the methods of gathenng subjective piobabihty estimates, and bettei methods aie being devised tor peifoiming sensitivity
analysis
One aspect of investment decisions still eludes the capabilities of this technique
This is the pioblem ot sequential decision making—that is, the analysis ot a
numbei ot highly lntei 1 elated investment decisions oeeuiimg at Jiffeient points
B-248
RICHARD F
USE
OF
DECISION
FOR
IIESP0S
TREE
TO
A NEW
AND P A U L A
ANALYZE
PRODUCT
STKASSMANN
INVESTMENT
ALTERNATIVES
INTRODUCTION
NPV
(Milhons of
A Review of Decision Trees
The decision tiee approach, a technique very similar to dynamic progiammmg,
is a convenient method foi lepresenting and analyzing a series of investment
decisions to be made ovei time (see Figuie 3) Each decision point is repiesented
by a numbered square at a foik or node in the decision tree Each branch extending from a fork repiesents one of the alternatives that can be chosen at this
decision point At the fust decision point the two alternatives 111 the example
shown in Eiguie 3 aie "introduce pioduct nationally" and "introduce product
regionally " (It is assumed at this point that the decision has aheady been made
to introduce the product in some way )
In addition to representing management decision points, decision trees represent chance events The forks in the tree where chance events influence the
outcome aie indicated by cucles The chance event forks or nodes rn the example
repiesent the various levels of demand that may appear for the product
A node representing a chance event geneially has a piobabihty associated with
each of the blanches emanating from that node This probability is the likelihood
that the chance event will assume the value assigned to the particular branch
The total of such probabilities leadmg from a node must equal 1 In our example,
the probability of achieving a large demand in the regional introduction of the
pioduct is 0 7, shown at the branch leading from node A Each combination of
decisions and chance events has some outcome (in this case, net present value, or
NPV) associated with it
T H E A N A L Y S IS OF I N V E S T M E N T
JB-249
DECISIONS
FIGURE 4
NET
P R E S E N T V A L U E OF I N V E S T M E N T
ALTERNATIVES
F O R A NEW P R O D U C T I N T R O D U C T I O N
ALTERNATIVE
CHANCE
EVENT
PROBABILITY
OF
CHANCE EVENT
NET
PRESENT
VALUE
$
7 5
Lorge national demand
5
L a r g e r e g i o n a l , limited
national demand
2
1 0
Limited demand
3
-4 0
Introduce product r e g i o n a l l y
L a r g e n a t i o n a l demand
5
4 5
(and distribute
nationally
if regional demand is
large)
L a r g e r e g i o n a l , limited
Limited demand
3
1 0
I n t r o d u c e product r e g i o n a l l y
L a r g e national demand
5
2 5
2
70
3
1 0
I n t r o d u c e product n a t i o n o l l y
(and do not
nationally)
distribute
national demand
2
-0
5
L a r g e r e g i o n a l , limited
n a t i o n a l d e m a nd
Limited demand
EXPECTED
NPV
I
>
Î 2 75
)
j,
2 44
)
>
)
1 95
The optimal sequence of decisions in a decision tiee is found by stalling at the
ught-hand side and "rolling backward " At each node, an expected NPV must
be calculated If the node is a chance event node, the expected N P V is calculated
for all of the branches emanating fiom that node If the node is a decision point,
the expected N P V is calculated for each bianch emanating fiom that node, and
the highest is selected In eithei case, the expected NPV of that node is earned
back to the next chance event or decision point by multiplying it by the probabilities associated with blanches that it tiavels ovei
Thus in Figuie 3 the expected N P V of all branches emanating fiom chance
event node C is $3 05 million ($4 5 X 71 + $ - 0 5 X 29) Similarly, the expected NPV at node D is $2 355 million Now "rolling back" to the next node—
decision point 2—it can be seen that the alternative with the highest NPV is
"distribute nationally," with an NPV of $3 05 million This means that, if the
decision makei is evei confionted with the decision at node 2, he will choose to
distubute nationally, and will expect an NPV of $3 05 million In all further
analysis he can ignore the other decision bianch emanating from node 2 and all
nodes and branches that it may lead to
To peiform further analysis, it is now necessary to c a n y this NPV backwaid
in the tiee The branches emanating from chance event node A have an overall
expected NPV of $2 435 million ($1 X 0 3 + $3 05 X 0 7) Similaily, the expected
NPV at node B is 2 75 million These computations, summanzed in Figure 4,
show that the alternative that maximizes expected NPV of the entne decision
tiee is "introduce nationally" at decision point 1 (Note that m this particular
case theie aie no subsequent decisions to be made )
One drawback of the decision tiee approach is that computations can quickly
become unwieldy The number of end points on the decision tiee increases veiy
lapidly as the number of decision points oi chance events increases To make this
appioach practical, it is necessaiy to limit the number of branches emanating
fiom chance event nodes to a veiy small number This means that the probability
B-250
RICHARD
IIESP0S
F
AND
PAUL
A
STKASSMANN
distribution of chance events at each node must be lepiesented by a veiyy few
point estimates
As a result, the ausweis obtained fiom a decision tiee analysis are often inadequate The single answei obtained (say, net piesent value) is usually close to
FIGURE 5
RANGE
FOR
PROBABILITY
5 r
M INTRODUCE
EACH
OF
POSSIBLE
OF
THREE
OUTCOMES
ALTERNATIVES
NATIONALLY
EXPECTED
NPV =
»2 75
(b)
INTRODUCE
REGIONALLY
THEN ACT
"OPTIMALLY""
%
1
E X P E CTED
NP V »2 44
1
J
$-4
1
-
1
2
|
1
0
f ,
1
1
JJ
2
N P V (Millions ol Dollars)
"Meaning, in this case, to maximize expected NPV
PROBABILITY
(c) I N T R O D U C E
ONLY
EXPECTED
NPV =
REGIONALLY
1
A
i
—
;
i
i
1
«
\r
T H E ANALYSIS OF INVESTMENT
JB-251
DECISIONS
the expectation of the probability distnbution of all possible N P V s However, it
may vary somewhat from the expected N P V , depending on how the point estimates were selected fiom the undeilying distributions and on the sensitivity of
the N P V to this selection piocess Fuithennoie, the decision tiee appioach gives
no information on the lange of possible outcomes fiom the investment or the
probabilities associated with those outcomes This can be a serious d i a w b a c k
I n the example in Figuies 3 and 4, the decision tiee appioach indicated t h a t
mtiodueing the pioduct nationally at once would be the optimal strategy foi
maximizing expected N P V Ilowevei, the N P V of $2 75 million is simply the
mean of thiee possible values of NPV, which aie themselves lepiesentative of an
entire lange of possible values, as shown in Figuie 5a Conipaiing the iange of
N P V s possible under each possible set of decisions shows a vastly difteient view
of the outcome (See Figuies 5b and 5c )
Although the hrst alternative has the highest expected N P V , a lational managei could easily piefer one of the othei two The choice would depend on the
utility function or the aveision to lisk of the managei oi his organization A managei with a linear utility function would choose the fh&t alternative, as shown m
Figure Ga Howevei, it is piobably tiue that moU manageis would not choose the
first alternative because of the high chance of loss, and the highei utility value
t h a t they would assign to a loss, as shown in Figuie bb This conseivatism in
management is, to a large extent, the result of the system of rewards and punishments t h a t exists in m a n y laige corpoiations today Whether it is good oi bad is
a complex question, not discussed heie
I n spite of these shoi tconnngs, the decision tiee appioach is a veiy useful analytical tool I t is paiticularly useful for conceptualizing investment planning and
foi controlling and monitoiing an investment t h a t stretches out over time F o r
these reasons, the decision tree appioach has been, and will continue to be an
important tool for the analysis of investment decisions
FIGURE 6
EXAMPLES
OF
UTILITY
FUNCTIONS
(b)
iol
LINEAR UTILITY
MORE
FUNCTION
NONLINEAR
VALUEOFs
•Li »
0
TYPICAL
UTILITY
V A L U E OF
Com
CHANGE IN AS5ET5 ($)
-L<
•ss
0
FUNCTION
f
Go,,,-
change in a s s e t s ($)
B-252
RICHARD F I I E S P 0 S A N D P A U L A
STKASSMANN
Combining These Appioaches: Stochastic Decision Tiees
The complementary advantages and disadvantages of risk analysis and decision tiees suggest that a new technique might be developed that would combine
the good points of each and eliminate the disadvantages The concept of stochastic decision tiees, introduced in the remainder of this article, is intended to be
such a combination
ty
i
The stochastic decision tree approach is similar to the conventional decision
tree approach, except that it also has the following featuies.
% All quantities and factors, including chance events, can be represented by
continuous, empuieal piobability distnbutions
The infoimation about the lesults from any or all possible combinations of
decisions made at sequential points in time can be obtained in a probabilistic
fonn
If The piobability distnbution of possible lesults fiom any paiticular combination of decisions can be analyzed using the concepts of utility and risk
A discussion of each of these features follows
Replacement of Chance Event Nodes by Pi obabihty
Distributions
The mclusion of piobability distributions for the values associated with chance
events is analogous to addmg an aibitianly laige number of branches at each
chance event node In a conventional decision tiee, the addition of a large number
of blanches can serve to lepresent any empirical piobability distnbution Thus
m the pievious example, chance event node B can be made to approximate more
closely the desiied continuous piobability distnbution by incieasing the number
of blanches, as shown in Figure 7a and 7b However, this approach makes the
tiee very complex, and computation very quickly becomes burdensome or impractical Therefore, two 01 three branches are usually used as a coaise approximation of the actual continuous piobability distnbution
Since the stochastic decision tree is to be based on simulation, it is not necessary
to add a great many blanches at the chance event nodes In fact, it is possible to
reduce the number of branches at the chance event nodes to one (See Figure 7c )
Thus, in effect, the chance event node can be eliminated Instead, at the point
wheie the chance event node occurred, a random selection is made on each iteration fiom the appiopnate probabilistic economic model such as the break-even
ehait shown in Figuie 8 and the value selected is used to calculate the N P V for
that particular iteration The single biancli emanating fiom this simplified node
then extends onward to the next management decision point, or to the end of the
tiee This results in a diastic stieainlimng of the decision tiee as illustrated m
Figuie 9
Replacement oj All Specific Values by Probability
Distributions
In a conventional decision tree, factois such as the size of the investment in a
new plant facility aie often assigned specific values Usually these values are expressed as single numbers, even though these numbeis aie often not known with
cei tainty
B-254
RICHARD F
IIESP0S
AND PAUL A
STKASSMANN
FIGURE 8
TYPICAL
USED
TO
SELECT
PROBABILISTIC
VALUES
OF
ECONOMIC
FACTORS
AT
MODEL
CHANCE
EVENT
NODES
If the values of these factors could be represented instead by probability distributions, the degiee of uncertainty characterizing each value could be expressed
The stochastic decision tree appioach makes it possible to do this Since the approach is basically a simulation, any 01 all specific values in the investment analysis can be lepresented by probability distributions On each iteration in the
simulation, a value for each lactor is randomly selected from the appiopriate
frequency di&tnbution and used in the computation Thus, m the example, NPV
can be calculated fiom not only empincal distributions of demand, but also probabilistic estimates of investment, cost, price, and othei factors
Evaluating all Possible Combinations oj Decisions
Since this stochastic decision tree appioach greatly simplifies the structure of
the decision tiee, it is often possible to evaluate by complete enumeration all of
the possible paths through the tree Poi example, if there aie five sequential decisions in an analysis and each decision offeis two alternatives, theie aie at most
32 possible paths through the decision tree This number of paths is quite manageable computationally And since most decision points aie two-sided ("build"
oi "don't build," foi example), oi at worst have a veiy small nuinbei of alternatives, it is often feasible and convenient to evaluate all possible paths through a
decision tiee when the stochastic decision tree appioach is used
Why is it sometimes desnable to evaluate all possible paths through a decision
tree? As the inquiiy into the lisk analysis appioach showed, decisions cannot
always be made conectly solely on the basis of a single expected value for each
tactoi The loll-back technique ot the conventional decision tiee necessarily deals
T H E A N A L Y S I S OF I N V E S T M E N T
DECISIONS
JB-255
FIGURE 9
SIMPLIFIED
DECISION
TREE
DISTRIBUTE
NATIONALLY
DO N O T
•DISTRIBUTE
NATIONALLY
A
only with expected values It evaluates decisions (moie exactly, sets of decisions)
by comparing their expectations and selects the largest as the best, in all cases
Howevei, the stochastic decision tiee approach pioduces pi obabilistic lesults
for each possible set of decisions These probability distributions, associated with
each possible path thiough the decision tree, can be compared on the basis of
their expectations alone, if this is consideied to be sufficient But alternative sets
of decisions can also be evaluated by comparing the piobabihty distributions associated with each set of decisions, in a mannei exactly analogous to usk analysis
(The details of this technique aie discussed in the next section ) Thus, the stochastic decision tiee appioach makes it possible to evaluate a series of interielated
decisions spiead ovei time by the same kinds of risk and uncertainty c a t e n a that
one would use in a conventional ask analysis
In a laige decision tree pioblem, even with the simplifications affoided by the
stochastic decision tiee approach, complete enumeiation of all possible paths
through the tiee could become computationally unpractical, or the compaason
of the piobabihty distabutions associated with all possible paths might be too
laboaous and costly
In such a case, two simplifications aie possible Fust, a modified veision of the
loll-back technique might be used This modified loll-back would take account
of the piobabihstic natuie of the information being handled Blanches of the tiee
would be eliminated on the basis of dominance rathei than simply expected
value [71 For example, a blanch could be eliminated if it had both a lower expected letuin and a lugliei vaaance than an alternative branch A number of
possible sets of decisions could be eliminated this way without being completely
evaluated, leaving an efficient set of decision sequences to evaluate 111 moie detail
Computation could also be 1 educed by making decision mles befoie the simulation, such that if, on any iteiation, the value of a chance event exceeds some critenon, the lesulting decision would not be consideied at all This has been done
B-256
RICHARD F
IIESP0S
A N D PAUL A
STKASSMANN
FIGURE 10
THE
GPSS
CONCEPT
WITH
RISK
OF
DECISION
TREES
SIMULATION
m the example shown in Figure 3 If a limited demand appears at node A, national
mtroduction of the product will not be evaluated In the sunulation, if demand
were below some specified value, the simulation would not proceed to the decision
point 2 This technique only saves computation effort—it does not simphfy the
structure of the tiee, and if the cntenon is chosen propeily, it will not affect the
final outcome
Recording Results in the Foi m of Probability
Distributions
I t has aheady been shown that probability distiibutions are moie useful than
single numbeis as measures ol the value of a paiticular set of decisions The simulation approach to the analysis permits one to get these probability distiibutions
relatively easily I t is tiue that the method smacks of biute foice However, the
biute force lequired is entuely on the pait of the computei and not at all on the
part of the analyst
The technique is sunply this On each iteration or path through the decision
tree, when the computer encounters a binary decision point node, it is instructed
to "split itself in two" and perform the appropriate calculations along both branches of the tree emanating fiom the decision node (The same logic apphes to a
node with three oi more branches emanating fiom i t ) Thus, when the computer
completes a single iteiation, an NPV will have been calculated for each possible
path thiough the decision tree These NPVs are accumulated in sepai ate probability distiibutions This simulation concept is lllustiated in Figuie 10
At the completion of a suitable number of iterations, there will be a probability
distubution of the N P V associated with each set of decisions that it is possible to
make in passing thiough the tree These diffeient sets of decisions can then be
T H E A N A L Y S I S OF I N V E S T M E N T
DECISIONS
JB-257
compared, one against the other, m the usual lisk analysis mattei, as if they were
alternative investment decisions (which in fact they aie) T h a t is, they can be
compared by taking into account not only the expected return, but also the shape
of each probability distnbution and the effects of utility and risk On the basis of
this, one can select the single best set of decisions, 01 a small numbei of possibly
FIGURE 11
RESULTS
STOCHASTIC
PROBABILITY
OF
DECISION-TREE
ANALYSIS
B-258
RICHARD F. HESPOS A N D P A U L A.
STRASSMANN
acceptable sets. These sets of sequential decisions can then be evaluated and a
decision whether or not to undertake the investment can be made by comparing
it to alternative investments elsewhere in the corporation or against alternative
uses for the money.
Ail
Example
To illustrate the kinds of results that can be expected from a stochastic decision tree analysis, the new product introduction problem described earlier has
been solved using this method. The results are shown in Figure 11.
The differences in the expected values of the outcomes can now be seen in
proper perspective, since the results show the relationship of the expected values
to the entire distribution of possible outcomes. Moreover, the expected values of
these distributions will not necessarily be identical with expectations resulting
from the conventional decision tree approach, because:
1. The interdependencics among the variables were not accounted for by the
conventional approach.
2. The small number of point estimates used to approximate an enLire distribution under the conventional approach did not utilize all the available information.
With the three alternatives presented in this form, it is easier to understand
why a rational manager might choose an alternative other than the one with the
highest expected value. Presented with the full range of possible outcomes related
to each alternative, he can select that alternative most consistent with his personal utility and willingness to take risk.
Using the Stochastic Decision Tree Approach
Stochastic decision trees described here combine the best features of both risk
analysis and conventional decision trees and are actually simpler to construct
and use than either of these. The steps for collecting data and conceptualizing
the problem are the same for the stochastic decision tree approach as they are for
the risk analysis approach. These steps are:
1. Gather subjective probability estimates of the appropriate factors affecting
the investment.
2. Define and describe any significant inlerdependenciea among factors.
3. Specify the probable timing of future sequential investment decisions to be
made.
4. Specify the model to be used to evaluate the investment.
The stochastic decision tree approach is ideally suited to the computer language
known as General Purpose Systems Simulator (GPSS). [5, 14] Although this
language is not now capable of handling very complex interdependencies without
certain modifications, it permits the solution of a very wide range of investment
problems.
The structuring and solving of several sample problems have indicated that the
stochastic decision tree approach is both easy to use and useful. The example in
Figures 4, 5 and 6 shows emphatically how the stochastic decision tree approach
T H E ANALYSIS OF INVESTMENT
can detect
and display
DECISIONS
JB-259
t h e p r o b a b l e o u t c o m e s of a n i n v e s t m e n t s t r a t e g y
that
would be d e e m e d optimal by the conventional decision tree approach, b u t
that
m a n y m a n a g e m e n t s w o u l d d e f i n i t e l y r e g a r d a s u n d e s i r a b l e . O t h e r w o r k is b e i n g
done on b o t h sample problems a n d real world problems, a n d on the
and standardization
development
( t o a l i m i t e d e x t e n t ) of t h e c o m p u t e r p r o g r a m s f o r p e r f o r m -
ing this analysis.
Summary
T h e s t o c h a s t i c d e c i s i o n t r e e a p p r o a c h t o a n a l y z i n g i n v e s t m e n t d e c i s i o n s ¡8 a n
e v o l u t i o n a r y i m p r o v e m e n t o v e r p r e v i o u s m e t h o d s of a n a l y z i n g i n v e s t m e n t s .
combines
t h e a d v a n t a g e s of s e v e r a l e a r l i e r a p p r o a c h e s ,
a d v a n t a g e s , a n d is e a s i e r to
eliminates several
It
dis-
apply.
References
1. ANDERSON, S. L. AND HAI OUT, H. G . , "A T w o - b y - T w o D e c i s i o n P r o b l e m , "
Chemical
Engineering
Progress,
Vol. 57, N o . 5, M a y 1961.
2. ANTHONY, RODERT N . ( E d i t o r ) , Papers on Return
on Investment,
Harvard business
S c h o o l , B o s t o n , 1959.
3. " C h a n c e F a c t o r s M e a n i n g a n d U s e , " A t l a n t i c R e f i n i n g C o m p a n y , P r o d u c i n g D e p a r t m e n t , J u l y 1962.
4. DEAN, J OEL, Capital Budgeting,
N e w Y o r k , C o l u m b i a U n i v e r s i t y P r e s s , 1951.
5. GOUUON, G . , " A G e n e r a l P u r p o s e S y s t e m s S i m u l a t o r , " IBM Systems
Journal,
Vol. I.,
S e p t e m b e r 1962.
6. HERTZ, DAVI D B. , " R i s k A n a l y s i s in C a p i t a l I n v e s t m e n t , " Harvard
Business
Review,
J a n u a r y - F e b r u a r y , 1961.
7. HESS, SI DNEY W . AND Q UIGLEY, HARRY A., " A n a l y s i s of R i s k in I n v e s t m e n t s U s i n g
M o n t e Carlo T e c h n i q u e , " C h e m i c a l E n g i n e e r i n g P r o g r e s s S y m p o s i u m Series N o . 42,
Vol. 59.
8. HI LLI ER, FREDERI CK, S., S t a n f o r d U n i v e r s i t y , " T h e D e r i v a t i o n of P r o b a b i l i s t i c I n f o r m a t i o n for t h e E v a l u a t i o n of R i s k y I n v e s t m e n t s , " Management
Science,
April,
1963.
9. MAQEE, J OHN F. , " D e c i s i o n T r e e s for D e c i s i o n M a k i n g , " Harvurd
Business
Review,
J u l y - A u g u s t , 1964.
19. MACEE, J OHN F. , " H o w to U s e D e c i s i o n T r e e s in C a p i t a l I n v e s t m e n t , " Harvard
Business Review, S e t p e m b e r - O c t o b e r , 1961.
11. MARKOWI TZ, HARRY, Portfolio
Selection,
Efficient
Diversification
of Investments,
New
Y o r k , J o h n W i l e y a n d S o n s , 1959.
12. MASSE, PI ERRE, Optimal
Investment
Decisions,
P r e n t i c e H a l l , 1962.
13. MCLEAN, J OHN G . , " HOW t o E v a l u a t e N e w C a p i t a l i n v e s t m e n t s , " Harvard
Business
Review, N o v e m b e r - D e c e m b e r , 1958.
14. R e f e r e n c e M a n u a l G e n e r a l P u r p o s e S y s t e m s S i m u l a t o r I I , I B M , 1963.
15. SCHLAI EER, ROBERT, Probability
and Statistics
for Business
Decisions,
McGraw-Hill,
1959.
16. TERDOROH, GEOROB, Business
Investment
Policy,
Machinery and Allied Products
I n s t i t u t e , W a s h i n g t o n , D . O . , 1958.
17. TI I ORNB, H. C. AND WI SE, D. C., A m e r i c a n Oil C o m p a n y , " C o m p u t e r s in E c o n o m i c
E v a l u a t i o n , " Chemical
Engineering,
April 29, 1963.
18. " V e n t u r e A n a l y s i s , " C h e m i c a l E n g i n e e r i n g P r o g r e s s T e c h n i c a l M a n u a l , A m e r i c a n
I n s t i t u t e of C h e m i c a l E n g i n e e r s .