Evalua tion of Research Expen dit u res Using Trian g ular Dis tri bu tion Functions and Monte Carlo Methods FRANK B. SPROW
Eualuating the potential pr0J;tability of research effort has long been subject to the vagaries of estimating optimism and pessimism. A more substantial basis for project evaluation has been devised and the author illustrates its use onventional methods for the evaluation and comC parison of new research expenditures generally incorporate only one estimate each cash flow associated of
with the project-that is, each expense and income item. Then the sensitivity of the analysis to the factors underlying the cash flows is tested. Unfortunately, these estimates may be clouded by varying degrees of optimism and pessimism, especially since the result of increasing or decreasing an estimate is usually easily foreseen. In this work the estimates are cast in terms of probability distributions, forcing the estimator to consider both optimistic and pessimistic cases. Before discussing the scheme in detail, we must first establish the basic mechanism of the analysis, including : -The
method of handling the expenditures and returns resulting from the research program. Also, the recognition of criteria by which alternative investment proposals may be judged. -Formulation of the probabilistic framework for the economic estimates-i.e., the choice of distribution functions for individual cash flows. -A method for analysis of the network of cash flows which consists of the research program and its commercial implementation.
METHOD OF TREATING INDIVIDUAL CASH FLOWS The method of discounted cash flow has been discussed by many authors [see, for example, (7) ] and is generally accepted as being among the best available for the evaluation of new investment opportunities. This method consists of discounting the cash flows according to the time period in which they occur. For example, an immediate return on a research program is worth more than an equal return delayed by several years during which the research dollars spent have not yielded any financial return. In this procedure each cash flow
is discounted at an interest rate which is specified at the outset of the analysis. The criterion of investment value is taken in this work to be the present worth of the program-that is, the sum of all the discounted cash flows evaluated at a particular interest rate. Other criteria may also be informative, such as the discounted cash flow return (that interest rate which produces a present worth of zero). The present worth method has been chosen for this study because the cash flows themselves are directly indicated. The mathematical tools discussed are applicable to any investment criterion or cash flow scheme. Of course, all cash flows should be evaluated on an after-tax basis during the discounting procedure.
SPEC1FICATION OF PROBABI LlTY DISTRIBUTION FUNCTIONS Once we have decided to cast each economic estimate in terms of a complete probability distribution rather than a single estimate or some combination of multiple estimates, a particular distribution function must be chosen. While risk analysis can be made on the basis of distribution functions obtained from subjective probability estimates (3, 4) there is little reason to believe that estimation is accurate enough to assign the individual probabilities associated with cash flows. In addition, no analytical function has been convincingly shown to represent a realistic model for the distribution of economic events. Given then that no distribution has a priori justification for use in these analyses, we seek a function which possesses certain desirable characteristics : -The function should contain a certain number of parameters to be specified by the economic estimator. These parameters should be ones with which most estimators are comfortable-i.e., actual cash flows rather than probability assignments. It would be desirable if the distribution function were completely defined by the economic estimates. VOL 59
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/
1
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0
0.2
0.6
04
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v
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x
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Figwr 2. Cumuiati~cprobability functim fa triangular distribution d h m = 0.75 '
-The function should be capable of beimg skewed by the estimates-that is; the probability distribution should not be restricted to symmetry about a mean value. -The distribution should be amenable to mathemati& analysis, such as imcombination with other cash flows. Although numerous distribution functions could no doubt be formulated which satisfy the above criteria, a skpple triangular distribution (Figure 1) has proved suitable for several analyses. The economic estimates to be made are of the minimum, most liiely (peak of the distribution), and maxinium cash flows asgociated with each segment of the research program and its commercial implementation. For example; each research segment is e x p r e s d in three forms: the m i n i u m cost (the simplest scheme is successful, everything works the 6rst time); the most likely cost (the costs associated with the research program elements considered most probable) ; the miximum cost (a pessi&stic estimate, complex solutions to the technological problems involved are necessary). Similarly, the economic advantages of the commercialization of the research project are expressed as pessimistic, *st likely, and optinhtic estimates. Another distribution function utilizing these three estimates is the beta distribution used in the well known PERT treatment for the scheduling of networks of events. As has been pointed out by MacCrimmon and Ryavec (6), the PERT beta distributions are not completely specified by these tlpx estimates as the triangular distribution;. the PERT treatment also requires ,the specification of the standard deviation of the distribution in terms of its bounds. Grubb (2) has also cast. some doubts about the assumptions and. statistics of the PERT prkedure. We have used both the PERT beta distribution and the triangular distribution in several analyses, and see no advantage in the beta; the tri36
INDUSTRIAL AND ENGINEERING CHEMISTRY
angular distribution is much more amenable to analyses of sequences of events. Figure 1b illustrates the normalization of the triangular distribution function shown in Figure l a on the interval [O,l] of the distributed variable x. That is, x represents a fraction of the range of allowable cash flows. The normalization requires that
l'f(x)dx = 1 The probability density function f ( x ) is then given by
where m is the value of x when the distribution function is at its peak. The probability of the distributed variable having a value of x or greater is given by
F(x) = 1
*" -m
for x
5m
(1 (1
- m)
for x
2m
F(x) =
X)P
(3)
The limits of F(x) are 0 and 1. Figure 2 illustrates F(x) as a function of x for a value of m = 0.75.
SOLVING FOR THE OVERALL PRESENT WORTH DISTRIBUTION Next a formalism for the combination of the distribution functions for each cash flow should be established. For a large number of events the analytical solutions for even fairly simple cash flow networks become unwieldy.
For certain situations in which a large number of cash flows are to be summed into a present worth, the central limit theorem yields the result that the distribution of present worth is normally distributed with a mean value equal to the sum of individual cash flow means, and a variance equal to the sum of individual variances. This proves a useful concept for the simplest schemes in which only one path containing a large number of events connects the initiation of research to the commercialization of a process. Unfortunately, in many analyses this requirement is not satisfied. MacCrimmon and Ryavec (6) have recently presented an interesting discussion of the analysis of networks, and a similar study has been made by Van Slyke ( 8 ) , who used Monte Carlo methods to analyze a PERT network based on several distribution functions. Hess and Quigley (4) and Hertz (3) have suggested Monte Carlo procedures to arrive at an overall distribution function for a number of distributed economic variables. In the Monte Carlo technique a value of F ( x ) is randomly selected from the distribution of cash flows for each stage of the research program. The value of x is then calculated from this value (for the triangular distribution, Equation 3 would be used). Since x is the fractional range of cash flow, the cash flow corresponding to x is:
+
Cash flow = minimum cash flow x(maximum cash flow - minimum cash flow)
promising new catalyst system (the costs associated with this predevelopmental program are not considered in the following analysis). I t appears that the new catalyst will increase process yield significantly ; the research group estimates that yield credits will be from 3 (pessimistic) to 18 (optimistic) cents per barrel of feed to the commercial unit over the existing catalyst. For a 10,000-barrel-per-day feed rate to the commercial unit, the credits shown in Table I can be calculated. However, catalyst costs may increase and estimates of this debit are also given in Table I. Assuming that these results were obtained using a very small bench-scale reactor, a costly development program would probably be necessary before the new catalyst is used in the commercial reactor. Yearly research costs (pessimistic, most likely, and optimistic) are estimated for each year of a 5-year development program in Table I. Only one program is envisioned at a particular time. If other procedures, such as an independent engineering study to cut catalyst costs, are to proceed in parallel with the research program,
TABLE 1. Year
Low
1 2 3
150 120 160
4 5
ao
(4)
These cash flows calculated by the Monte Carlo procedure for each segment of the program are discounted after taking the appropriate tax factor into account and are then combined to yield a present worth. At this point one Monte Carlo trial has been completed. The procedure is then repeated, randomly selecting F ( x ) values for each distribution of cash flows and obtaining various resulting values for the present worth. A large number of trials yields an overall distribution function for the present worth of the research program. I t should be noted that the Monte Carlo procedure gives completely correct results only for the case where the individual events are independent. If the segments of a program are closely coupled, it is best to combine these segments into a single segment and distribution funpion before the Monte Carlo procedure is used.
EXAMPLE While the technique provides its most unique utility when applied to large networks of cash flows (such as a PERT network) in which parallel cash flows occur, a very simple example should serve to illustrate the types of input and output information involved in the analysis. Assume that a program of exploratory research in. catalysis has led to the preliminary development of a
AUTHOR Frank B. S’row is on the staf of the Baytown Petroleum Research Laboratory of Esso Research and Engineering Co., Baytown, Tex.
COST ESTIMATES FOR T H E EXAMPLE
Years
6-15
Research Costs, Most Likely
O$/bbl. fed 0 M$/yr.
Years
Low
6-15
3$/bbl. fed 110 M$/yr.
High
160 250 400 120 40
20 low
M$
190 380 640 480 80
Catalyst Cost Debits Most Likely
2.8$/bbl. fed 102 M$/yr. Yield Credits Most likely
lO#/bbl. fed 365 M$/yr.
High
6.2#/bbl. fed 226 M$/yr. High
18#/bbl. fed 657 M$/yr.
the distribution function for this study could be included in the Monte Carlo analysis. As stated earlier, the distribution functions associated with each yearly research segment have to be independent for the Monte Carlo procedure to apply. This in essence assumes that a good year can follow a bad year with little interdependence. Breakdown by year is convenient because it simplifies the discounting procedure; if, however, the yearly research segments are indicated to be closely coupled, appropriate independent segments should be considered. In our simple analysis of the example we assume that the yearly segments are indeed sufficiently independent. Distribution functions are then calculated for each program segment and the commercialization of the research program. For example Figure 1a illustrates VOL. 5 9
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the distribution of cash flows for the first year of the research program. The normalized distribution fu tion is shown in Figure l b , with the indicated mod 0.75 on x[O,l]. Figure 2 shows the curve of the c tive probability F(x) os. x for m = 0.75. Dis curves for the four remaining research years, the ca debit, and the yield credit are similarly obtained. A random number between 0 and 1 is then generate either by hand using a table of random digits (7)or on digital computer to yield one trial value of from which x and thus one trial cash flow can be cal lated. The cash flows (after discounting) for each of seven periods are combined to obtain the first Mo Carlo trial present worth. A process life of 10 year a discount rate of 15% have been adopted in thisex to reflect the normal business risks associated with th operation of the commercial unit. If the p m question involves unusual business risk, a highe count rate might be used. This is repeated a num of times, and the results of 500 computer trials in Figure 3. This histogram is easily transfor the dark solid curve shown in Figure 4. Two dent Monte Carlo procedures of 50 trials ea indicated. For fairly rough calculations, suffice, these smaller sets of trials are usually satisfactn The procedure using 50 trials can be done by ha with a table of random digits and desk calculator . few hours. Little advantage in accuracy is gener found by using more than 500 trials. The resulting graph can be quite informative. example, if only the most likely estimates were used present worth of 223 M$ would be calculated. use of the entire distribution generated by the t estimates for each cash flow yields a good deal mor information. For example, there is only a 42% cha of exceeding this 223 MS and, for example, only a 2 chance of exceeding a present worthof 300 M$. O n other hand, there is a 13% chance of obtaining a ne tive present worth for the research program. Ba on the entire distribution of events, the manager be able to make more meaningful decisions abo between alternative projects than with single est1 methods. Alternative treatments can be easily inserted into framework of the analysis to see their effect on present worth (or other criteria). distribution For example, the effect of a shorter research (in the example, less than 5 years) could be test Normal tests of the sensitivity of the analysis to the inpu variables can also be made.
(a,
-!I
-2““
-,XI
iw
0 100 300 4w 500 W K t l WORIH, TnOUSANDS OF DOLLARS
”””
Figure 3. Histogram gentrated from 500 Matte Cmlo rriols
-lw
0
100
203
3w
4w
% MK) I
WWNT W O R M THOUSANDS Oi DOLLAM
Figwe 4. Owdl dirtribulionof p e d w t h
REFERENCES (1) Bicrman, E., Smidf, S., “The CQW BudanYark, 1966.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Dcbdon.’’ Maomillan, Nnr
