Risk Analysis in Chemical Plant Investment E. L. Reynard nvestment analysis has been given considerable thought in recent years. Most current articles center on ways to take into account the time value of money. But the use of this concept suffers from one main problem-present value and internal rate of return calculations use single-valued estimates of gross revenue and cost which imply certainty. Granted, high risk can be expressed in present value calculations by using a discounting factor corresponding to a high interest rate (8). However, judgment has to be exercised on the final revenue figure as a whole. In this method, judgment cannot be reasonably broken down and applied to the factors contributing to net revenue. In recent years the concept of decision trees has been introduced to cope with the problem of future uncertainties. In this concept, the array of alternatives is displayed. For every alternative decision, best judgments are applied to the probabilities associated with the events that could result from each decision. This sequence is carried as far into the future as desired. When future outcomes are evaluated, one can work back to the present to select the immediate alternative investment that is most suitable (5, 6, 9). The calculations are somewhat cumbersome and by necessity use single-valued probability estimates. As the real world of continuous uncertainty is incorporated (3, 4, computers are required to simulate probability distributions describing uncertainty in the relevant variables. This last reference (4) in easence incorporates all of the good features (discounting and sequential decisions) and eliminates most of the bad features (single-valued estimates) of all of the literature on investment analysis. But computer simulation of probability distributions presupposes good knowledge of the shape of the probability distributions. While time value of capital considerations are important, there are four points that are perhaps more so.
I
Conaidor Malor Unce~iaintioaConMbuting to Profit Unceriainty
When one is determining profitability of a future investment, uncertainty comes from two major sources, the economic environment and investment estimates. The economic environment may influence profitability in many ways. For purposes of illustration, attention will be confined to product price and product volume. We must freely admit that the future of price and volume is indeed uncertain. This in turn leads to reasoning that some probability distribution exists, but is it practical to speculate whether the distribution is skewed to the right or left? I think a simple guess that the normal distribution applies is as far as our reasoning can usually stretch. The other area of profitability uncertainty considered in this illustration is in the investment estimate. Again, some probability distribution describes the uncertainty
ANNUAL PROFIT
.Effort is better spent analyzing risk. .Uncertainty of many factors influences profit. .There is a limit on the precision with which we can
.
define uncertainty. We need a relatively simple approximation to ascertain the profit uncertainty caused by the combined influence of contributing uncertainties.
Figure 1. Uncntnintier in indimdual projectiani cause m e r t u n t y profif projection V O L 5 8 NO. 7 JULY 1 9 6 6
61
SENSITIVITY ANALYSIS OF THREE FACTORS VS. COMBINED UNCERTAINTY In each case the potential variability of certam key factors involved with B chemical plant investment is recognized. Using the best information
Opfimirfic Ertimofe Price Annual volume Plant investment Working capital Variable cost Total investment
Pessimistic Erfimdlr
Best Guess
3O#/lb.
35#/lb. 9,000,000 Ib. $2,000,000 $200,000 10#/lb. $2,200,000
401/lb. 12,000,000 Ib. $1,500,000 $200,000 $1,700,000
6,000,000 Ib. $2,500,000 t200,000 $2,700,000
SENSITIVITY ANALYSIS APPROACH
Using the parameters for chemical X, we first find manufacturing costs: Annual volume
-
Variable costs (raw mat'ls) Fixed costs A. Operating labor B. Control lab C. Overhead D. Annual strai ht line dcpreciatian a t IO%, of plant investment E. Taxes & insurance a t 2.5% of plant investment F. Maintenance a t 7.570 of plan1 investment Total fixed costs Total mfg. costs
$2200
~
_
__
$1700
s2200 -
~
t2700
1200
1200
1200
!NO
900
900
600
600
600
250 50 250
250 50 250
250 50 250
250 50 250
250 50 250
250 50 250
250 50 250
250 50 250
250 250
150
200
250
150
200
250
150
200
250
50
37.5
50
62.5
37.5
50
62.5
37.5
50
62.5
112.5 850 $2050
150 950 $2150
187.5 1050 82250
112.5 850 $1750
150 950 $1850
187.5 1050 51950
112.5 850 $1450
150 950 $1550
187.5 1050 $1650
~
a
6,000,000 Ih.
_ .
$1700
Total investments
9,000,000 Ib. I $2700 $1700 12200 $2700 _ __ -
12,000,000 Ib.
.
-
All dollaifigmts in rhournnds of dollorr
Ned,find annual profits for each price: Annual volume Total investment-
9,000,000 Ib.
12,000,000 Ib. $1700
I
12200
I
$2700
Sales Mfg. costs
4800 2050
4800 2150
Pretax profit Net profit at 48% tax Return an in".
2750 1430
2650 1378 63%
2550 1326 49%
84%
4800 2250
__-
4200 2050
4200 2150
4200
Mfg. wsts Pretax profit Nct profit at 48% tax Return on in".
2150 1118 69 %
2050 1066 48%
1950 1014
Sales Mfg. costs
3600 2050
3600 2150
3600 2250
Pretax profit Net profit at 48% tax Return on in".
1550 806 47 %
1450 754 34%
1350 702 26%
Sales
2250
38%
81700
$2200
3600 3600 1750 1850 ~1850 1750 962 910 57% 41 %
6,000,000 lb. 82700
$1700
$2200
$2700
3600 1950
2400 1450
2400 1550
2400 1650
1650 858 32%
3150 1750
3150 1850
3150 1950
1400 728 43%
1300 676 31 %
1200 624 23y0
950 494 29%
2100 1450 650 338 20%
--
850 442 20%
2100 1550 550 286 13%
750 390 14%
2100 1650 450 234 9%
-
A2
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
2700 1750 950 494 29%
I 2700 1850 850 442 20%
2700 1950 750 390 14%
1800 1450 350 182 11%
1800 1550 250 130 6%
1800 1650 150
78 3%
INVOLVING THREE FACTORS available, the following table is developed for chemical X:
about the estimate, but the exact nature is unknown. Approximating the probability with the normal distribution will be adequate for our purposes.
P ROPQSEO STAT I ST I CA L APPROACH
Use Familiar Statistical Concepts
Reduce computation by using the relationships developed in this article. Assume the same parameters as before.
We do not need the age of computers to determine an encompassing measure of risk. Several concepts are combined to produce the needed result. Standard deviations are common parlance in the field, of quality control. For our purposes, a standard deviation is a convenient way to view investment risk. Standard deviation of profit nicely defines the interval which has about a two thirds chance of enclosing expected profit. In a loose statistical sense, we can say that annual profit has about a two thirds chance of falling in the interval defined as: expected annual profit =k standard deviation of profit. The question that remains is how do we estimate this interval by some practical means?
Annual profit
=
(sales volume) (price/lb.) [fixed costs (variable costs) (sales volume)]
+
Using the best guess parameters, we calculate annual profit as follows : Pretax profit = (9,000,000 Ib.) ($0.35/1b.) [$950,000 ($O.lO/lb.) (9,000,000 lb.)] = $3,150,000 - [950,000 900,000] = $1,300,000
+
After tax profit = $1,300,000 = $676,000
+
- (0.48 ($1,300,000)
Annual return on total investment = $676,000/2,200,000 == 3170
The variability is calculated by using the following derived relationship: best guess
) )’
Convert Accounting Factors into Algebraic Expressions
- variable cost
optimistic volume
- pessimistic volume
optimistic price
- pessimistic
(-
2
price
6 maint. % of plant inv.
(best guess)’ volume
+
6
> +
tax
%
- of+plantinv. 100
depreciation yo Of plant inv.
optimistic (plant inv.
6,000,000 Ib.
6
)’ + (9,000,000 lb.)’ X
($0.40/lb. - $0.30/lb. 6 $1,500,000
- $2,500,000)z]1/2 6
u Pretax profit
E
u After tax profit
fixed production cost
estimated variable cost unit >x best guess of volume
(
E
$293,000 $152,000
Similarly, there is about a ‘/3 chance that return on total investment lies between: 3524,000 ____
22,200,000
and
$828,000 ~
c2,200,000
or
24%
and
38%
(1)
+
+
+
operating labor control laboratory chemists overhead depreciation at A70 of plant investment taxes and insurance at B7G of plant investment maintenance material and labor at Cyo of plant investment
+
+
Thus, expected annual profit equals : (best guess of price/unit) operating labor chemists
- (0.48) ($293,000)
Thus, there is about a 2/3 chance that after tax profit lies between $524,000 and $828,000
)]
For estimating purposes, certain fixed production costs might be based on plant investment. For example, annual fixed production costs might equal :
$293,000 E
+
(
plant inv.
Substituting we have:
-
Expected annual profit = Expected sales - expected cost of production income = (best guess of price/unit) (best guess of volume) -
- pessimistic)~]1” 6
12,000,000 Ib.
The first step is to relate our profit estimate algebraically to the factors that comprise it (2). Thus, an estimate could be of the following nature:
(”
+l&+
best guess of
+ control laboratory + overhead
‘) (best guess
of plant investment)
estimated variable cost unit
)
+
best guess (of volume)]
(2)
E. L. Reynard i s Administrative Specialist at Staufer Chemical Co. in Richmond, Calif., with his professional emkhasis currently on Budgetary and project analysis. He was trained as a chemical engineer and has worked in process design, cost analysis, and operations research. The comments and suggestions of Ramsey G. Campbell, Marlin Brnnett, John F. Heil, and Guy W . Roy are gratefully acknowledged. AUTHOR
VOL. 5 0
NO.
7
JULY
1966
63
optimistic price
Relate Profit Uncertainty to Economic Uncertainties
Next, we need to relate uncertainty in the economic and estimating variables to uncertainty in the dependent variable, profit. To do this we borrow a seldom used statistical approximation ( 7 ) .
)+
pessimistic price
(-A
-H 100
c
pessimistic optimistic plant investment plant investment 6
(11)
Reflection
where Y
=f(X,
W , Z ) ; cr2Y is the variance of Y.
Applying this statistical relationship to o w situation, we get: ’
dprofit
(E) volume ”+
profit
02
d profit (d
5)price -+
d profit b plant investment)
UZ
cr2
plant investment
(4)
The partial differentials are easily found by differentiating the profit relationship (Equation 2). b_ profit _ __- (best guess of price/unit) d volume (estimated variable cost/unit)
a- profit a price
(5)
- best guess of volume profit b plant investment
(7)
Turning to the problem of estimating the variance of these three important variables, we must borrow a page from the literature on PERT (Program Evaluation and Review Technique) (7). Using the same simplifying assumptions that guided the PERT calculations for ,the successful Polaris Project, we get the following analogs.
u2 volume e
2 price
u2 plant investment
=
(
optimistic volume
pessimistic volume
)
optimistic price
2
(8)
pessimistic price (9)
optimistic pessimistic plant investment - plant investment estimate estimate 6
Hence, we finally get the standard deviation of annual profit by substituting Equations 5 through 10 into Equation 4.
cr
profit
(
[(
f :
optimistic volume
64
price unit pessimistic)* volume
+(
X
best )2 guess of volume
X
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
We now have a fairly convenient means of analyzing investment risk without a computer. Thc ,jiidgments needed are the same ones that wotilcl Iw i i s c ~ li n any reasonably thorough analysis. Onc coiiltl t ; i k ( . the conventional sensitivity analysis approach a n d c.saiiiinc all 27 combinations of optimistic, best estiiiiatc, and pessimistic values for these three illustrated variables. But besides being an overwhelming amount of data, extreme cases of all-optimistic or all-pessimistic variables are very unlikely. Other combinations have varying degrees of likelihood. I t is more realistic to use the single value for profit standard deviation which gives effect to the likely uncertainty of all variables. The proposed method easily permits inclusion of more economic factors. A thoroiigh sensitivity analysis involving four variables would require calculation of 81 combinations. The proposed method would add one more quantity to the profit standard deviation equation. An example is presented to illustrate the computational differences between the proposed statistical approach and conventional sensitivity analysis. The examples use the rate of return on investment (ROI) project evaluation criterion for simplicity’s sake. Other project evaluation criteria involving annual cash flows and time value of money could conceptually use an annual profit uncertainty as input. Conclusion
Basically, this paper discusses two points. The uncertainties of economic and investment factors need to be individually analyzed and then combined into a single, useful figure-overall profit uncertainty. The basic input of each factor’s uncertainty can be derived from optimistic, best guess, and pessimistic estimates. REFERENCES (1) Bennett, ( 2 . A , , Franklin, N. L., “Statistical Analysis in Chemistry and thc Chemical Industry,” Wiley. New York, 1954 (2) Churchman, C. I V . , k k o f l , R. L., Arnofl, E. L., “Introduction t o Opcrations Research,” Wiley, New York, 1957. (3) Hertz D . B “Risk Analysis in Capital Investment,” Harirard Business Reciew 42 (11, b5 (19z4). (4) Hespos, R . F.. Sirassman P . A. “Stochastic Decision Trees for the Analysis of Investment Decision,’’ Mnn’agernen; Science 11 ( i o ) , August (1965). ( 5 ) Magee, J. F., “Decision Trees for Decision Making,” Horuard Business Reuiew 4 2 (4), 126 (1964). ( 6 ) M a g e e J. F “ H o w to Use Decision Trees in Capital Investment,” Harward Business heuiea’k2 (5), 79 (1964). ( 7 ) PERT-Summary Report, Phase 1 , Special Projects Office, Bureau of Naval LVeapons, Department of Navy, FVashington, D. C., 1 9 5 8 . (8) Quinn, J. B., “ H o w L O Evaluate Research Output,” H o w a r d Business Reciew 38 (Z), 69 (1960). (9) Schlaifler, Robert, “Probability a n d Statistics for Business Decisions,” McGrawHill, New York, 1 9 5 9 .
