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A practical approach to risk and sensitivity analyses
Long
12
Range
Planning
Vol. 13
February
1980
A Practical Approach to Risk and Sensitivity Analyses Surendra S. Singhvi”
The financial evaluation of capital investment projects is made using one or more of the conventional techniques such as payback period, accountant’s rate of return, net present value and discounted cash flow rate of return. These techniques are generally used by firms to provide one ‘most likely’ estimate of the project’s performance expressed in terms of its pay back period, rate of return or net present value. This approach fails to take into account the extent of the uncertainty or risk associated with the project.’ This paper presents the nature and scope of risk and sensitivity analyses. A case study on the application of risk and sensitivity techniques is also included. .
What
is Risk?
1. Degree Project
A
0
1000 i 500 _’ 1000 1500
Probability
1 .oo 0.25 0.50 0.25 1 .oo
Definition
definition
of risk Per cent of total responses
of risk
Probability of not achieving a target return Variation in returns Payback period uncertain Uncertain market potential Entering an inexperienced area Success ratio (potential gain/potential loss) Miscellaneous
40 30 10 7 5 4 4
100
The project risk, in the survey, was defined by 10 per cent of the responding executives as the probability of not achieving a target return. Here, management is concerned only with negative variation. The second most significant definition of project risk according to 30 per cent of the respondents relates to variation in returns without regard to the direction of such variation. It is worth noting that only 10 per cent ofthe respondents define project risk in terms of the payback period.
Approaches
of risk illustrated Return
2. Management’s
Total
Risk, like beauty, lies in the eyes of the beholder. ‘Risk’ is generally defined as the estimated degree of uncertainty with rcspcct to the realization of expected future returns. The wider the range of expected future returns around the most likely estimate, the more risky is the investment. Accordingly, project B is more risky than project A in Table 1. Table
Table
Expected
to Cope with Risk
Project risk has been recognized by businesses of all sizes and types cithcr formally or intuitively. Various approaches are used by decision-makers to cope with the risk or uncertainty associated with a proposed investment such as :
return
1000 125 500 375 -
(4
1000
Requiring than the projects.
a lower payback period target payback period
for risky projects for normal risk
(b) Raising
the hurdle rate for risky projects in relation to the overall cost of capital. For example. if the overall cost of capital is 10 per cent, the hurdle rates and new business projccrs for cost saving, expansion, could be established at 10, 15 and 20 per cent respcctivcly.
According to one survey of IO9 industrial corporations,’ a question was asked to management, ‘What is meant when you say an invcstmcnt proposal is risky?’ Table 2 summarizes the management’s definition of risk.
‘Dr. Surendra S. Singhvi IS Assistant Treasurer of Armco Inc., Adjunct Professor of Finance at Miami University (U.S.A.).
and
(4
Reducing the best estimates of future for risky projects based on management
cash inflo\vs judgment.
A Practical
(4
(4
Using three-level estimates of DCF rate of returnoptimistic, most likely and pessimistic. A wider range of return indicates more risk for the project than a narrow range. One could assign probabilities to these estimates and calculate an expected rate of return as shown in Table 3. Applying a risk analysis using computer simulation model (Monte-Carlo) and the probability concept. This approach will be discussed later in detail.
Table
3. Three
level estimate
Approach
Table
to Risk and Sensitivity
5. Methods
No. of responses Changing required rate of return Adjusting cash flows probabilistic basis Adjusting cash flows subjective basis Changing required payback period Other methods
Estimates
Probability
Expected return (%)
Optimistic (high) Most likely (median) Pessimistic (low)
25 20 10
0.20 0.60 0.20
5 12 2
1 .oo
19
According to the survey of 109 industrial corporations, referred to earlier, the frequency of use made of various methods for analyzing the riskiness of a capital investment varies significantly. Table 4 summarizes the methodology for analyzing the riskiness of a capital investment. for analyzing
the riskiness
of a
Relative frequency of use
Payback Risk adjusted discount rate Measure expected variation in returns Simulation Certaintyequivalentapproach Other
Never (%)
Infrequently (%)
Frequently
Always
(%)
(%)
23
16
26
36
35
28
28
9
45 47
24 26
29 25
2 2
72 90
20 0
6 8
2 2
According to another survey of 234 firms,2 it was found that 71 per cent (166 firms) of the respondents explicitly accounted for risk in making capital budgeting decisions, and their methods of risk adjustment are summarized in Table 5.
What
Per cent of total
58
30
50
26
40
21
27 17
14 9
192
100
approach
DCF return (%)
Method
13
of risk adjustment
Total
Table 4. Methodology capital investment
Analyses
is ‘Risk-Analysis’?”
Risk analysis estimates a range of possible results of a proposed investment decision based on the given input data and states the probability that the overall result will be between specified ranges. It simulates the effects of the uncertainty surrounding key variables entering into the evaluation on the returns one is likely to achieve.
Risk analysis adds two distinctive features to the conventional one-figure measures of evaluation : first, it quantifies the uncertainty in the factors affecting the outcome of an investment decision. Second, it combines these estimates using a simulation process to produce a description of the risk attached to the proposed investment.
Quantifying
Uncertainty
Quantifying uncertainty is the first distinctive feature of risk analysis. Management selects those factors affecting cash flow whose uncertainty is to be quantified, i.e. capital investment amount, sales price, volume, working capital and the like. Judgment must be used in selecting only those factors which have significant effect on the project’s future cash flow. The appropriate members of management obtain the necessary data for each factor by (a) assigning the highest and lowest possible values for the factor, (b) splitting this range of values into 4-5 subranges, and (c) estimating the probability of the values lying in each subrange. Using these guidelines, a probability distribution of sales volume is illustrated in Table 6. Table
6. Quantifying
Sales volume (million tons/year) 5-l 0 IO-15 15-20 20-25 25-30
uncertainty
Probability of sales to be equal to or less Probability of sales than upper limit of in a given range the range shown 0.10 0.30 0.40 0.15 0.05 1
Development
illustrated
0.10 0.40 0.80 0.95 1 .oo
.oo
of Probabilities
Probability data should be developed by a group of experts representing various functions such as marketing, operations and the like. Each engineering, finance, member of the group should assign probabilities based on his own background and experience. An average of
14
Long Range
Planning
Vol. 13
February
these individual probabilities would express the group’s overall degree of confidence for the occurrence of an event. In some cases, an empirical evidence can be used to support the probabilities assigned. For example, the probability of base cost per ton could be assigned using the historical data as shown in Table 7.
Table
7. Empirically
1980 Table
Monthly sales of product X (s)
No. of months these costs were realized
Probability values
90-l 00 80-90 70-80 60-70
21 30 6 3
0.35 0‘50 0.10 0.05
60
1 .oo
illustrated
Deviation from mean
60 70 80 90 100
based probabilities
Base cost per ton ($1
9. Large variation
-20 -10 0 +10 +20
400 =
400 + 5 = $80.
Standard Deviation = -\/lo00
Development of probabilities and ranges could be improved through the so-called ‘interrogation technique’. For example, a group of 20 individuals was asked to estimate Bombay’s population. Using the interrogation technique, the population estimates for Bombay were developed and are summarized in Table 8.
10. Small variation
+ 5 = $14.
illustrated
Monthly sales of product Y (5)
Deviation from mean
Squared deviation
70 75 80 85 90
-10 -5 0 +5 +10
100 25 0 25 100
400
Table
8. Probability
Population estimate in millions
and interrogation
technique
Mean
250 =
400 7 5 = $80.
Standard Deviation = 4250 No. of participants
20
7 5 = $7.
Probability 0.00 0.05 0.05 0.10 0.35 0.35 0.10 0.00
50 and above 20-50 15-19 10-14 5-9 2-4 l-2 Less than 1
400 100 0 100 400 1000
Mean
Table
Squared deviation
1 .oo
All 20 participants agreed that the population of Bombay is somewhere between I and 50 million. The weighted average population came to 7.45 million, which is very the technique can close to the actual figure. Similarly, be used in estimating sales volume, sales price and other variables for a proposed investment.
Based on the standard deviation and mean values, one can estimate an interval or range showing the degree of confidence one has in such estimation. Statisticians have shown mathematically that about 68 per cent of sample observations normally lie between & 1 standard deviation from the mean, about 95 per cent observations lie between -12 standard deviations and 98 per cent observations lit between i2.33 standard deviations. These values can be found in any standard textbook on statistics. At the 98 per cent level, the confidence interval for the product Y example, discussed above, would be:
= ( smZZ)
’
(standa%lZiation)
’ (2’33)
= 80 z!z 7(2.33)
Statistical Techniques
= 563.7 to S96.3
Used
The standard deviation is a statistical technique to measure deviation from the mean. As a rule, the smaller the standard deviation value, the lower the variation or uncertainty. Table 9 provides an example of large variation or uncertainty related to a forecast of monthly sales rcvcnue of product X. An example of small variation or uncertainty forecast of monthly sales revenue of product in Table 10.
related to Y is shown
This means the product Y sales for any month between $63.7 and S96.3, and this estimation true 98 out of 100 times.
would should
be be
Using the standard deviation, Ibbotson and Sinquefield quantified the degree, of risk associated with common stocks, long-term government bonds, long-term corporate bonds, and U.S. Treasury Bills.4 Returns and risks for these securities for 1926-l 974 period are shown in Table 11. The highest return investment shows the
A Practical
11. Standard
deviation
Type of investment
risk
Arithmetic mean of annual returns (%)
Standard deviation of annual returns (%I
10.9
22.5
3.4
5.4
3.7 2.3
5.1 2.1
Common stock Long-term government bonds Long-term corporate bonds U.S. treasury bills
Computer
to measure
Figure 1 shows a risk profile chart prepared by the computer, which indicates returns on the horizontal axis and the probability of achieving these returns on the vertical axis. Point A on the chart can be interpreted as 99 per cent probability that the DCF return would be 17.6 per cent or higher, point B as 50 per cent probability that the DCF return would be 19.4 per cent or higher, point C as 16 per cent probability that the DCF return would be 20.4 per cent or higher, and point D as 1 per cent probability that the DCF return would be 21.3 per cent or higher.
Simulating the outcome is the second distinctive feature of risk analysis. Simulation calculations are made by a computer, which simulates the many possible outcomes of an investment decision. During simulation the computer chooses values at random from the probability distribution of each factor affecting the future cash flows. The computer then uses these random values to calculate the cash flow and DCF return over the project’s life. Then it repeats these calculations several times, each time choosing another set of values at random and generatingdifferent values of the DCF return.
Similarly, based on 200 trials, the computer estimated a mean payback period of 6.68 years with a standard deviation of 0.19 years. It can now be inferred that 98 out of 100 times, the payback period will fall between 6.24 and 7.12 years.
illustration
Normally, the data shown in Table 12.A are developed for economic justification study related to a real-life investment proposal. Using the conventional approach to project evaluation, the information shown in Table 12.B is provided to management for decisionmaking. So far no attempt is being made to quantify the degree of risk. Additional data as shown in Table 12.C
Table
12A. Risk analysis-an
illustration-Input
Capital expenditure (s)
Additional working capital (s)
0 1 2 3 4 5 6 7 8 9 10 11 12
1800 406
550 550 293 128 74
Total
2206
Year
‘Represents
undepreciated
15
Analyses
Based on 200 trials, the computer estimated a mean DCF rate of return of 19.4 per cent with a standard deviation of 0.8 per cent. It can now be inferred that 98 out of 100 times, the DCF return will fall between 17.6 and 21.3 per cent. However, it must be kept in mind that such estimation is valid to the extent our assumptions are valid.
Simulation”
Risk analysis-an
to Risk and Sensitivity
are required on a proposed project in order to apply the risk analysis technique. Basically, Table 12.C required the range of values with probability distribution for key variables entering into the cash flow projections. In this example, only six variables with three subranges for each variable have been used; however, the computer program can accommodate additional variables with additional subranges.
largest degree of risk and the vice-versa. The standard deviation, therefore, seems to be a reasonable measure of risk.
Table
Approach
1595
Sensitivity
Analysis
This analysis supplements the risk analysis applied to new projects. The objective is to determine the degree of change in terms of the project’s DCF return or net present value if the key variables entering into the financial justification are exposed to unfavourable
Data
Sales ($1 2500 5000 6333 6917 7250 7250 7250 7250 7250 7250 7250 7250 78,750
Manufacturing profit as per cent of sales 17.5 24.5 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0
General overhead as per cent of sales 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7
Residual value ($1
Depreciation (s)
1960’
58 133 150 150 150 150 150 150 150 150 150 150 150
1960
1841
book value of fixed assets plus recovery of 100 per cent working capital. Assumes no investment tax credit.
16 Table
Long Range
Planning
12B. Conventional
Year
approach
Capital expenditure (s)
0 1 2 3 4 5 6 7 8 9 10 11 12 Total
Vol. 13 to project
1980
evaluation
Additional working capital (s)
(1800) (406)
(2206)
February
Net income (s)
Residual value (s)
Net cash flow (s)
397 519 567 594 594 594 594 594 594 594 594
58 133 150 150 150 150 150 150 150 150 150 150 150
1960
376 589 670 744 744 744 744 744 744 2704
6346
1841
1960
6346
(550)
111
$;:; (128) (74)
(1595)
Depreciation W
(1742) (712)
(3
DCF return = 18.3 per cent. Payback period = 6.1 years.
Table
12C. Risk analysis-probability
input
Key variables
Deviation from median estimate (%)
Probability
Capital Expenditure +5 0 -2
High Median Low
0.1 5 0.82 0.03 1 .oo
Residual Value +1 0 -10
High Median Low
0.10 0.70 0.20 1 .oo
Sales forecast +5 0 -10
High Median Low
0.55 0.25 0.20 1 .oo
Additional working capital as per cent of sales High Median Low
+3 0 -10
0.25 0.25 0.50
+3 0 -2
0.10 0.30 0.60
General overhead per cent of sales High Median Low
1 .oo Manufacturing High Median Low
profit as per cent of sales +10 0 -5
0.60 0.30 0.10 1 .oo
.
A Practical
17.6%
18.3%
@
Approach
Conventional
to Risk and Sensitivity
DCF
Analyses
17
Return
1.00
\ m
--
0.80
0 6 0.60
I -------____
-r ---_
t
_--------
I
0.40
I II I II r- -----
0.20 1
I I L
O.OOb
..-.
-dT DCF- Return ,%
Figure
1. Risk profile
chart
variances. Effect on the DCF return of an assumed per cent unfavourable variance in each key variable the above illustration is shown in Table 13.
Table
13. Sensitivity
analysis
Original DCF return
19.4% 16.9 17.6 18.3 18.5 18.6 19.3
Manufacturing profit per cent of sales Sales revenue Capital expenditures General overhead per cent of Sales Working capital Residual value
10 in
If manufacturing profit as a per cent of sales is off by 10 per cent from the median value, and other things being equal, the DCF return will deteriorate from 19.4 to 16.9 per cent as shown in Table 13. In this example, the most sensitive variable appears to be the manufacturing profit as a percent of sales. One could have used I5 or 20 per cent unfavorable variance to test the sensitivity of key variables. Additional sensitivity analysis can be undertaken to see the effect of two or more variables on the DCF return. For example, if sales are off by 25 per cent and the manufacturing profit percent is off by 10 per cent, the resulting DCF return would be 12.9 per cent.
Capital Expenditure
as % of Sales
mm
10
DCF Return,
Figure
2. Graphic
presentation
of sensitivity
analysis
%
18
Long Range
Planning
Vol. 13
February
1980
lo-
9-
”
High Risk
8-
F.
/
6-
J 0
E.
D / ,*
8 Expected
F J A A
.
Figure
3. Risk policy
is better & B are Et E are B Ii are
1
10
12
DCF
Return,
%
than B & C better than C better than J better than G
considerations
Based on sensitivity .analyses, management can investigate and research thoroughly only those variables which are very sensitive in terms of their effect on DCF return, and need not spend any time on those variables which are least sensitive. The results of sensitivity analysis can be shown graphically to make them more understandable.6 For example, it can be seen in Figure 2 that manufacturing profit as per cent of sales is the most sensitive variable and the residual value is the least sensitive variable. Those variables whose curves are most flat (i.e. horizontal) arc most sensitive to change; conversely, those variables whose curves are most steep (i.e. vertical) are least sensitive to change.
Comparing Analysis
Moderate Risk
/I I
Risk 4
/
Projects with Risk
The use of risk analysis in comparing projects is shown in Table 14.
two
Table
14. Risk analysis
Amount of investment Life of investment Mean average DCF return Standard Deviation 98 per cent confidence interval
in comparing
projects
Project A
Project B
s2m 12 years 10% 1.7%
s2m 12 years 13% 6.9%
6-14%
-3-29%
Figure 3 emphasizes some risk policy considerations in between risk and selecting new projects.7 Trade-offs return are favorable to the decision-maker if the projects on the left of the curve (northwest corner) are rejected. For example, project F is better than projects B and C, since all three projects represent the same degree of risk as measured by standard deviation, but project F has the highest return on investment compared to B and C. The shape of the curve is dependent upon the decision-maker’s risk preference.
or more
Project B, other things being equal, is more profitable than project A, but it is also more risky. It must be remembered that high risk projects are not necessarily undesirable. What a firm needs is a balanced portfolio of investment projects representing high, medium and low risk.
A Critical Look at ‘Risk
Analysis’
According to some critics, the computerized-simulation based risk analysis as a management tool has not been successful despite its popularity after Hertz wrote his now ‘classic’ article in the January-February, 1964 issue of Harvard B14sirwrr ~wirw. In a study on the experiences
A Practical four major oil companies have had in using risk analysis, E. Eugene Carter has tried to separate those factors which have helped or hindered successful implementinherent in any ation. * Carter found few difficulties application of risk analysis in capital investments such as obtaining various assessments of risks and probabilities from individual managers, trading off risk against return in a structured way and adjusting procedures for evaluating and controlling projects for some of the after effects of risk analysis. He suggests that the degree of success with risk analysis depends on the resolution of several questions such as:
(9 How
mature each company
(ii) What (iii) How
is the risk analysis decides to use it?
is the origin is it fitted
of the decision
in with
company
technique to adopt
generated
and
(vi) What role does top management in deciding to install risk analysis?
it?
organization?
responsible (iv) How are the managers, technique prepared for handling it? (v) How are the data model form?
when
for using
put
together
reserve
the
in
Approach
to Risk and Sensitivity
analysis of risks, substituting living with risks as they arise.
Analyses
instead
These findings should be used with are based on a very small sample.
better
caution
19 means
of
since these
The key problem in implementing risk analysis seems to be the development of probability input and the selection of relevant variables for which the risk is to be quantified. Like any other technique’s successful implementation, top management must support the use of it and the people must be educated about its use. It is possible that an individual manager could misuse the risk analysis technique by using optimistic data and by quantifying irrelevant risks but these are not the shortcomings of the technique per se. It seems that the risk and sensitivity analyses tools could be used in evaluating major capital investment proposals to provide additional useful information to decision makers which would not be available to them otherwise. The author has used the tool in numerous situations and found it to be beneficial in most situations.
for itself
K. Larry Hastie suggests in an article that risk analysis has failed to improve decision making, and that the sensitivity analysis is the key to improving the communication of uncertainty to decision makers.” His concern is that the businessman is working with the same forecasting uncertainty or bias in risk analysis as that used to develop assumptions for single point estimates. In many cases, events such as the oil embargo occur that would be extremely difficult to expect and assess, yet they have significant impacts upon the project’s the optimism of managers profitability. In addition, affects the determinations of probabilities in risk analysis in the same way assumptions are determined for single point estimates. William K. Hall, based on his in depth studies of four large manufacturing firms and interviews with executives of I2 large firms, concludes that risk analysis is not having and probably will not have a measurable, positive impact on the planning process.1° Hall identifies two problem areas in implementing risk analysis: first, the decision as to who should quantify uncertainty and how this should be done; and second, the decision as to what uncertainties should be quantified. According to him, it is likely that managers will give up the formal
References
(1) J. William Petty, David F. Scott and Monroe M. Bird, The capital expenditure decision-making process of large corporations, The Engineering Economist, 20,166, Spring (1975)
(2) Glenn H. Petry, Effective use of capital budgeting tools, Business Horizon, p. 64, October (1975). (3)
David B. Hertz, Risk analysis in capital investment, Harvard Business Review, LVII, 169-I 81, September-October (1979).
(4)
R. G. lbbotson and R. A. Sinquefield, Stocks, bonds, bills and inflation : year-by-year historical returns, The Journal of Business, p. 40, January (1976).
(5)
An analyst can use a canned program for risk analysis offered by several time-sharing firms such as Corn-Share Inc., of Ann Arbor, Michigan, U.S.A.
(6)
Michael C. Luecke, Computer models : black box or management oriented ? Management Adviser, pp. 22-23, JanuaryFebruary (1973).
(7)
David B. Hertz, Investment policies that pay off, Harvard Business Review, pp. 96-l 08, January-February (1968).
(8)
E. Eugene Carter, What are the risks in risk analysis? Harvard Business Review, pp. 72-82, July-August (1972).
(9)
K. Larry Hastie, One businessman’s view of capital budgeting, Financial Management, p. 39, Winter (1974).
(IO)
William K. Hall, Why risk analysis isn’t working, Long Range Planning, 8, 25-29, December (1975).
(11)
Surendra S. Singhvi, Planning for Capifal Investments (Oxford, Ohio, U.S.A., Planning Executive Institute), pp. 5368 (1979).
12
Range
Planning
Vol. 13
February
1980
A Practical Approach to Risk and Sensitivity Analyses Surendra S. Singhvi”
The financial evaluation of capital investment projects is made using one or more of the conventional techniques such as payback period, accountant’s rate of return, net present value and discounted cash flow rate of return. These techniques are generally used by firms to provide one ‘most likely’ estimate of the project’s performance expressed in terms of its pay back period, rate of return or net present value. This approach fails to take into account the extent of the uncertainty or risk associated with the project.’ This paper presents the nature and scope of risk and sensitivity analyses. A case study on the application of risk and sensitivity techniques is also included. .
What
is Risk?
1. Degree Project
A
0
1000 i 500 _’ 1000 1500
Probability
1 .oo 0.25 0.50 0.25 1 .oo
Definition
definition
of risk Per cent of total responses
of risk
Probability of not achieving a target return Variation in returns Payback period uncertain Uncertain market potential Entering an inexperienced area Success ratio (potential gain/potential loss) Miscellaneous
40 30 10 7 5 4 4
100
The project risk, in the survey, was defined by 10 per cent of the responding executives as the probability of not achieving a target return. Here, management is concerned only with negative variation. The second most significant definition of project risk according to 30 per cent of the respondents relates to variation in returns without regard to the direction of such variation. It is worth noting that only 10 per cent ofthe respondents define project risk in terms of the payback period.
Approaches
of risk illustrated Return
2. Management’s
Total
Risk, like beauty, lies in the eyes of the beholder. ‘Risk’ is generally defined as the estimated degree of uncertainty with rcspcct to the realization of expected future returns. The wider the range of expected future returns around the most likely estimate, the more risky is the investment. Accordingly, project B is more risky than project A in Table 1. Table
Table
Expected
to Cope with Risk
Project risk has been recognized by businesses of all sizes and types cithcr formally or intuitively. Various approaches are used by decision-makers to cope with the risk or uncertainty associated with a proposed investment such as :
return
1000 125 500 375 -
(4
1000
Requiring than the projects.
a lower payback period target payback period
for risky projects for normal risk
(b) Raising
the hurdle rate for risky projects in relation to the overall cost of capital. For example. if the overall cost of capital is 10 per cent, the hurdle rates and new business projccrs for cost saving, expansion, could be established at 10, 15 and 20 per cent respcctivcly.
According to one survey of IO9 industrial corporations,’ a question was asked to management, ‘What is meant when you say an invcstmcnt proposal is risky?’ Table 2 summarizes the management’s definition of risk.
‘Dr. Surendra S. Singhvi IS Assistant Treasurer of Armco Inc., Adjunct Professor of Finance at Miami University (U.S.A.).
and
(4
Reducing the best estimates of future for risky projects based on management
cash inflo\vs judgment.
A Practical
(4
(4
Using three-level estimates of DCF rate of returnoptimistic, most likely and pessimistic. A wider range of return indicates more risk for the project than a narrow range. One could assign probabilities to these estimates and calculate an expected rate of return as shown in Table 3. Applying a risk analysis using computer simulation model (Monte-Carlo) and the probability concept. This approach will be discussed later in detail.
Table
3. Three
level estimate
Approach
Table
to Risk and Sensitivity
5. Methods
No. of responses Changing required rate of return Adjusting cash flows probabilistic basis Adjusting cash flows subjective basis Changing required payback period Other methods
Estimates
Probability
Expected return (%)
Optimistic (high) Most likely (median) Pessimistic (low)
25 20 10
0.20 0.60 0.20
5 12 2
1 .oo
19
According to the survey of 109 industrial corporations, referred to earlier, the frequency of use made of various methods for analyzing the riskiness of a capital investment varies significantly. Table 4 summarizes the methodology for analyzing the riskiness of a capital investment. for analyzing
the riskiness
of a
Relative frequency of use
Payback Risk adjusted discount rate Measure expected variation in returns Simulation Certaintyequivalentapproach Other
Never (%)
Infrequently (%)
Frequently
Always
(%)
(%)
23
16
26
36
35
28
28
9
45 47
24 26
29 25
2 2
72 90
20 0
6 8
2 2
According to another survey of 234 firms,2 it was found that 71 per cent (166 firms) of the respondents explicitly accounted for risk in making capital budgeting decisions, and their methods of risk adjustment are summarized in Table 5.
What
Per cent of total
58
30
50
26
40
21
27 17
14 9
192
100
approach
DCF return (%)
Method
13
of risk adjustment
Total
Table 4. Methodology capital investment
Analyses
is ‘Risk-Analysis’?”
Risk analysis estimates a range of possible results of a proposed investment decision based on the given input data and states the probability that the overall result will be between specified ranges. It simulates the effects of the uncertainty surrounding key variables entering into the evaluation on the returns one is likely to achieve.
Risk analysis adds two distinctive features to the conventional one-figure measures of evaluation : first, it quantifies the uncertainty in the factors affecting the outcome of an investment decision. Second, it combines these estimates using a simulation process to produce a description of the risk attached to the proposed investment.
Quantifying
Uncertainty
Quantifying uncertainty is the first distinctive feature of risk analysis. Management selects those factors affecting cash flow whose uncertainty is to be quantified, i.e. capital investment amount, sales price, volume, working capital and the like. Judgment must be used in selecting only those factors which have significant effect on the project’s future cash flow. The appropriate members of management obtain the necessary data for each factor by (a) assigning the highest and lowest possible values for the factor, (b) splitting this range of values into 4-5 subranges, and (c) estimating the probability of the values lying in each subrange. Using these guidelines, a probability distribution of sales volume is illustrated in Table 6. Table
6. Quantifying
Sales volume (million tons/year) 5-l 0 IO-15 15-20 20-25 25-30
uncertainty
Probability of sales to be equal to or less Probability of sales than upper limit of in a given range the range shown 0.10 0.30 0.40 0.15 0.05 1
Development
illustrated
0.10 0.40 0.80 0.95 1 .oo
.oo
of Probabilities
Probability data should be developed by a group of experts representing various functions such as marketing, operations and the like. Each engineering, finance, member of the group should assign probabilities based on his own background and experience. An average of
14
Long Range
Planning
Vol. 13
February
these individual probabilities would express the group’s overall degree of confidence for the occurrence of an event. In some cases, an empirical evidence can be used to support the probabilities assigned. For example, the probability of base cost per ton could be assigned using the historical data as shown in Table 7.
Table
7. Empirically
1980 Table
Monthly sales of product X (s)
No. of months these costs were realized
Probability values
90-l 00 80-90 70-80 60-70
21 30 6 3
0.35 0‘50 0.10 0.05
60
1 .oo
illustrated
Deviation from mean
60 70 80 90 100
based probabilities
Base cost per ton ($1
9. Large variation
-20 -10 0 +10 +20
400 =
400 + 5 = $80.
Standard Deviation = -\/lo00
Development of probabilities and ranges could be improved through the so-called ‘interrogation technique’. For example, a group of 20 individuals was asked to estimate Bombay’s population. Using the interrogation technique, the population estimates for Bombay were developed and are summarized in Table 8.
10. Small variation
+ 5 = $14.
illustrated
Monthly sales of product Y (5)
Deviation from mean
Squared deviation
70 75 80 85 90
-10 -5 0 +5 +10
100 25 0 25 100
400
Table
8. Probability
Population estimate in millions
and interrogation
technique
Mean
250 =
400 7 5 = $80.
Standard Deviation = 4250 No. of participants
20
7 5 = $7.
Probability 0.00 0.05 0.05 0.10 0.35 0.35 0.10 0.00
50 and above 20-50 15-19 10-14 5-9 2-4 l-2 Less than 1
400 100 0 100 400 1000
Mean
Table
Squared deviation
1 .oo
All 20 participants agreed that the population of Bombay is somewhere between I and 50 million. The weighted average population came to 7.45 million, which is very the technique can close to the actual figure. Similarly, be used in estimating sales volume, sales price and other variables for a proposed investment.
Based on the standard deviation and mean values, one can estimate an interval or range showing the degree of confidence one has in such estimation. Statisticians have shown mathematically that about 68 per cent of sample observations normally lie between & 1 standard deviation from the mean, about 95 per cent observations lie between -12 standard deviations and 98 per cent observations lit between i2.33 standard deviations. These values can be found in any standard textbook on statistics. At the 98 per cent level, the confidence interval for the product Y example, discussed above, would be:
= ( smZZ)
’
(standa%lZiation)
’ (2’33)
= 80 z!z 7(2.33)
Statistical Techniques
= 563.7 to S96.3
Used
The standard deviation is a statistical technique to measure deviation from the mean. As a rule, the smaller the standard deviation value, the lower the variation or uncertainty. Table 9 provides an example of large variation or uncertainty related to a forecast of monthly sales rcvcnue of product X. An example of small variation or uncertainty forecast of monthly sales revenue of product in Table 10.
related to Y is shown
This means the product Y sales for any month between $63.7 and S96.3, and this estimation true 98 out of 100 times.
would should
be be
Using the standard deviation, Ibbotson and Sinquefield quantified the degree, of risk associated with common stocks, long-term government bonds, long-term corporate bonds, and U.S. Treasury Bills.4 Returns and risks for these securities for 1926-l 974 period are shown in Table 11. The highest return investment shows the
A Practical
11. Standard
deviation
Type of investment
risk
Arithmetic mean of annual returns (%)
Standard deviation of annual returns (%I
10.9
22.5
3.4
5.4
3.7 2.3
5.1 2.1
Common stock Long-term government bonds Long-term corporate bonds U.S. treasury bills
Computer
to measure
Figure 1 shows a risk profile chart prepared by the computer, which indicates returns on the horizontal axis and the probability of achieving these returns on the vertical axis. Point A on the chart can be interpreted as 99 per cent probability that the DCF return would be 17.6 per cent or higher, point B as 50 per cent probability that the DCF return would be 19.4 per cent or higher, point C as 16 per cent probability that the DCF return would be 20.4 per cent or higher, and point D as 1 per cent probability that the DCF return would be 21.3 per cent or higher.
Simulating the outcome is the second distinctive feature of risk analysis. Simulation calculations are made by a computer, which simulates the many possible outcomes of an investment decision. During simulation the computer chooses values at random from the probability distribution of each factor affecting the future cash flows. The computer then uses these random values to calculate the cash flow and DCF return over the project’s life. Then it repeats these calculations several times, each time choosing another set of values at random and generatingdifferent values of the DCF return.
Similarly, based on 200 trials, the computer estimated a mean payback period of 6.68 years with a standard deviation of 0.19 years. It can now be inferred that 98 out of 100 times, the payback period will fall between 6.24 and 7.12 years.
illustration
Normally, the data shown in Table 12.A are developed for economic justification study related to a real-life investment proposal. Using the conventional approach to project evaluation, the information shown in Table 12.B is provided to management for decisionmaking. So far no attempt is being made to quantify the degree of risk. Additional data as shown in Table 12.C
Table
12A. Risk analysis-an
illustration-Input
Capital expenditure (s)
Additional working capital (s)
0 1 2 3 4 5 6 7 8 9 10 11 12
1800 406
550 550 293 128 74
Total
2206
Year
‘Represents
undepreciated
15
Analyses
Based on 200 trials, the computer estimated a mean DCF rate of return of 19.4 per cent with a standard deviation of 0.8 per cent. It can now be inferred that 98 out of 100 times, the DCF return will fall between 17.6 and 21.3 per cent. However, it must be kept in mind that such estimation is valid to the extent our assumptions are valid.
Simulation”
Risk analysis-an
to Risk and Sensitivity
are required on a proposed project in order to apply the risk analysis technique. Basically, Table 12.C required the range of values with probability distribution for key variables entering into the cash flow projections. In this example, only six variables with three subranges for each variable have been used; however, the computer program can accommodate additional variables with additional subranges.
largest degree of risk and the vice-versa. The standard deviation, therefore, seems to be a reasonable measure of risk.
Table
Approach
1595
Sensitivity
Analysis
This analysis supplements the risk analysis applied to new projects. The objective is to determine the degree of change in terms of the project’s DCF return or net present value if the key variables entering into the financial justification are exposed to unfavourable
Data
Sales ($1 2500 5000 6333 6917 7250 7250 7250 7250 7250 7250 7250 7250 78,750
Manufacturing profit as per cent of sales 17.5 24.5 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0
General overhead as per cent of sales 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7
Residual value ($1
Depreciation (s)
1960’
58 133 150 150 150 150 150 150 150 150 150 150 150
1960
1841
book value of fixed assets plus recovery of 100 per cent working capital. Assumes no investment tax credit.
16 Table
Long Range
Planning
12B. Conventional
Year
approach
Capital expenditure (s)
0 1 2 3 4 5 6 7 8 9 10 11 12 Total
Vol. 13 to project
1980
evaluation
Additional working capital (s)
(1800) (406)
(2206)
February
Net income (s)
Residual value (s)
Net cash flow (s)
397 519 567 594 594 594 594 594 594 594 594
58 133 150 150 150 150 150 150 150 150 150 150 150
1960
376 589 670 744 744 744 744 744 744 2704
6346
1841
1960
6346
(550)
111
$;:; (128) (74)
(1595)
Depreciation W
(1742) (712)
(3
DCF return = 18.3 per cent. Payback period = 6.1 years.
Table
12C. Risk analysis-probability
input
Key variables
Deviation from median estimate (%)
Probability
Capital Expenditure +5 0 -2
High Median Low
0.1 5 0.82 0.03 1 .oo
Residual Value +1 0 -10
High Median Low
0.10 0.70 0.20 1 .oo
Sales forecast +5 0 -10
High Median Low
0.55 0.25 0.20 1 .oo
Additional working capital as per cent of sales High Median Low
+3 0 -10
0.25 0.25 0.50
+3 0 -2
0.10 0.30 0.60
General overhead per cent of sales High Median Low
1 .oo Manufacturing High Median Low
profit as per cent of sales +10 0 -5
0.60 0.30 0.10 1 .oo
.
A Practical
17.6%
18.3%
@
Approach
Conventional
to Risk and Sensitivity
DCF
Analyses
17
Return
1.00
\ m
--
0.80
0 6 0.60
I -------____
-r ---_
t
_--------
I
0.40
I II I II r- -----
0.20 1
I I L
O.OOb
..-.
-dT DCF- Return ,%
Figure
1. Risk profile
chart
variances. Effect on the DCF return of an assumed per cent unfavourable variance in each key variable the above illustration is shown in Table 13.
Table
13. Sensitivity
analysis
Original DCF return
19.4% 16.9 17.6 18.3 18.5 18.6 19.3
Manufacturing profit per cent of sales Sales revenue Capital expenditures General overhead per cent of Sales Working capital Residual value
10 in
If manufacturing profit as a per cent of sales is off by 10 per cent from the median value, and other things being equal, the DCF return will deteriorate from 19.4 to 16.9 per cent as shown in Table 13. In this example, the most sensitive variable appears to be the manufacturing profit as a percent of sales. One could have used I5 or 20 per cent unfavorable variance to test the sensitivity of key variables. Additional sensitivity analysis can be undertaken to see the effect of two or more variables on the DCF return. For example, if sales are off by 25 per cent and the manufacturing profit percent is off by 10 per cent, the resulting DCF return would be 12.9 per cent.
Capital Expenditure
as % of Sales
mm
10
DCF Return,
Figure
2. Graphic
presentation
of sensitivity
analysis
%
18
Long Range
Planning
Vol. 13
February
1980
lo-
9-
”
High Risk
8-
F.
/
6-
J 0
E.
D / ,*
8 Expected
F J A A
.
Figure
3. Risk policy
is better & B are Et E are B Ii are
1
10
12
DCF
Return,
%
than B & C better than C better than J better than G
considerations
Based on sensitivity .analyses, management can investigate and research thoroughly only those variables which are very sensitive in terms of their effect on DCF return, and need not spend any time on those variables which are least sensitive. The results of sensitivity analysis can be shown graphically to make them more understandable.6 For example, it can be seen in Figure 2 that manufacturing profit as per cent of sales is the most sensitive variable and the residual value is the least sensitive variable. Those variables whose curves are most flat (i.e. horizontal) arc most sensitive to change; conversely, those variables whose curves are most steep (i.e. vertical) are least sensitive to change.
Comparing Analysis
Moderate Risk
/I I
Risk 4
/
Projects with Risk
The use of risk analysis in comparing projects is shown in Table 14.
two
Table
14. Risk analysis
Amount of investment Life of investment Mean average DCF return Standard Deviation 98 per cent confidence interval
in comparing
projects
Project A
Project B
s2m 12 years 10% 1.7%
s2m 12 years 13% 6.9%
6-14%
-3-29%
Figure 3 emphasizes some risk policy considerations in between risk and selecting new projects.7 Trade-offs return are favorable to the decision-maker if the projects on the left of the curve (northwest corner) are rejected. For example, project F is better than projects B and C, since all three projects represent the same degree of risk as measured by standard deviation, but project F has the highest return on investment compared to B and C. The shape of the curve is dependent upon the decision-maker’s risk preference.
or more
Project B, other things being equal, is more profitable than project A, but it is also more risky. It must be remembered that high risk projects are not necessarily undesirable. What a firm needs is a balanced portfolio of investment projects representing high, medium and low risk.
A Critical Look at ‘Risk
Analysis’
According to some critics, the computerized-simulation based risk analysis as a management tool has not been successful despite its popularity after Hertz wrote his now ‘classic’ article in the January-February, 1964 issue of Harvard B14sirwrr ~wirw. In a study on the experiences
A Practical four major oil companies have had in using risk analysis, E. Eugene Carter has tried to separate those factors which have helped or hindered successful implementinherent in any ation. * Carter found few difficulties application of risk analysis in capital investments such as obtaining various assessments of risks and probabilities from individual managers, trading off risk against return in a structured way and adjusting procedures for evaluating and controlling projects for some of the after effects of risk analysis. He suggests that the degree of success with risk analysis depends on the resolution of several questions such as:
(9 How
mature each company
(ii) What (iii) How
is the risk analysis decides to use it?
is the origin is it fitted
of the decision
in with
company
technique to adopt
generated
and
(vi) What role does top management in deciding to install risk analysis?
it?
organization?
responsible (iv) How are the managers, technique prepared for handling it? (v) How are the data model form?
when
for using
put
together
reserve
the
in
Approach
to Risk and Sensitivity
analysis of risks, substituting living with risks as they arise.
Analyses
instead
These findings should be used with are based on a very small sample.
better
caution
19 means
of
since these
The key problem in implementing risk analysis seems to be the development of probability input and the selection of relevant variables for which the risk is to be quantified. Like any other technique’s successful implementation, top management must support the use of it and the people must be educated about its use. It is possible that an individual manager could misuse the risk analysis technique by using optimistic data and by quantifying irrelevant risks but these are not the shortcomings of the technique per se. It seems that the risk and sensitivity analyses tools could be used in evaluating major capital investment proposals to provide additional useful information to decision makers which would not be available to them otherwise. The author has used the tool in numerous situations and found it to be beneficial in most situations.
for itself
K. Larry Hastie suggests in an article that risk analysis has failed to improve decision making, and that the sensitivity analysis is the key to improving the communication of uncertainty to decision makers.” His concern is that the businessman is working with the same forecasting uncertainty or bias in risk analysis as that used to develop assumptions for single point estimates. In many cases, events such as the oil embargo occur that would be extremely difficult to expect and assess, yet they have significant impacts upon the project’s the optimism of managers profitability. In addition, affects the determinations of probabilities in risk analysis in the same way assumptions are determined for single point estimates. William K. Hall, based on his in depth studies of four large manufacturing firms and interviews with executives of I2 large firms, concludes that risk analysis is not having and probably will not have a measurable, positive impact on the planning process.1° Hall identifies two problem areas in implementing risk analysis: first, the decision as to who should quantify uncertainty and how this should be done; and second, the decision as to what uncertainties should be quantified. According to him, it is likely that managers will give up the formal
References
(1) J. William Petty, David F. Scott and Monroe M. Bird, The capital expenditure decision-making process of large corporations, The Engineering Economist, 20,166, Spring (1975)
(2) Glenn H. Petry, Effective use of capital budgeting tools, Business Horizon, p. 64, October (1975). (3)
David B. Hertz, Risk analysis in capital investment, Harvard Business Review, LVII, 169-I 81, September-October (1979).
(4)
R. G. lbbotson and R. A. Sinquefield, Stocks, bonds, bills and inflation : year-by-year historical returns, The Journal of Business, p. 40, January (1976).
(5)
An analyst can use a canned program for risk analysis offered by several time-sharing firms such as Corn-Share Inc., of Ann Arbor, Michigan, U.S.A.
(6)
Michael C. Luecke, Computer models : black box or management oriented ? Management Adviser, pp. 22-23, JanuaryFebruary (1973).
(7)
David B. Hertz, Investment policies that pay off, Harvard Business Review, pp. 96-l 08, January-February (1968).
(8)
E. Eugene Carter, What are the risks in risk analysis? Harvard Business Review, pp. 72-82, July-August (1972).
(9)
K. Larry Hastie, One businessman’s view of capital budgeting, Financial Management, p. 39, Winter (1974).
(IO)
William K. Hall, Why risk analysis isn’t working, Long Range Planning, 8, 25-29, December (1975).
(11)
Surendra S. Singhvi, Planning for Capifal Investments (Oxford, Ohio, U.S.A., Planning Executive Institute), pp. 5368 (1979).