In this era of advancing technology, successful managers need to make investment decisions that determine the future success of their companies by drawing systematically on the specialized knowledge, accumulated information, experience, and skills of many people. In previous lessons, the investment analyses were all considered to be made under "norisk" conditions. In this lesson, we add in the uncertainties when evaluating an energy/mining project. The objective of investment decisions is to invest available capital where we have the highest probability of generating the maximum possible future profit. The use of quantitative approaches to incorporate risk and uncertainty into analysis results may help us be more successful in achieving this objective over the long run.
At the successful completion of this lesson, students should:
This lesson will take us one week to complete. Please refer to the Course Syllabus for specific time frames and due dates. Specific directions for the assignment below can be found within this lesson.
Reading  Read Chapter 6 of the textbook. 

Assignment  Homework and Quiz 6. 
If you have any questions, please post them to our discussion forum, located under the Modules tab in Canvas. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.
So far, in previous lessons, effect of risk and uncertainty haven’t been considered in our economic evaluations and the analyses were assumed to be of norisk condition. In this case, the probability of success and achieving anticipated results is assumed to be 100%, but in reality, some degree of uncertainty is involved and this probability is much lower. The economic analyses that don’t include risk and uncertainty are based on “best guess,” and the results aren’t highly accurate and reliable. For example, if a study shows 20% and 25% ROR for project A and B, the manager would probably choose project B over A. But what if the probability of success is 90% for project A and 40% for project B? This example shows how important it is to consider the effect of risk and uncertainty as a component in economic evaluations.
Quantitative methods, along with informal analysis, are used for decision making under risk and uncertainty. Quantitative methods aim to provide the best possible set of information to decisionmakers so that they may apply their experience, intuition, and judgment to achieve the final decision; the decision that leads to maximum possible future profit with the highest probability. There are several different approaches that can be used to quantitatively incorporate risk and uncertainty into analyses. These include sensitivity analysis or probabilistic sensitivity analysis to account for uncertainty associated with possible variation in project parameters, and expected value or expected net present value or rate of return analysis to account for risk associated with a finite probability of failure. The use of sensitivity analysis is advocated for most economic analyses and the use of expected value analysis is advisable if a finite probability of project failure exists. Sensitivity analysis is a means of evaluating the effects of uncertainty on investment by determining how investment profitability varies as the parameters are varied that effect economic evaluation results.
Sensitivity analysis can show how results change if the input parameter changes. If we change one input parameter (such as initial investment) and the result (such as NPV of the project) varies significantly in a wide range, then we say the result is sensitive to the specified input parameter. Here, we aim to find the most sensitive variables. The input parameter investigated for sensitivity analysis usually includes initial investment, selling price, operating cost, project life, and salvage value. If probabilities of occurrence are associated with various levels of each investment parameter, sensitivity analysis becomes probabilistic sensitivity analysis.
It may now be evident to you that the term “uncertainty” as used in this lesson refers to possible variation in parameters that effect investment evaluation. “Risk” refers to the evaluation of an investment using a known mechanism that incorporates the probabilities of occurrence for success and failure and/or of different values of each investment parameter. Both uncertainty and risk influence almost all types of investment decisions, but especially investment involving research and development for any industry and exploration for minerals and oil or gas.
Please watch the following video (3:24): Risk, Uncertainty, and Sensitivity Analysis.
Italicized sections are from Stermole, F.J., Stermole, J.M. (2014) Economic Evaluation and Investment Decision Methods, 14 edition. Lakewood, Colorado: Investment Evaluations Co.
As explained before, in sensitivity analysis, we aim to discover the magnitude of change in one variable (here, output variables) with respect to change in other variables (here, input parameters). Then, we can rank the variables based on their sensitivity. It helps the decisionmaker to understand the parameters that have the biggest impact on the project.
The following example introduces a single variable sensitivity analysis. Please note that here we assume variables are independent and have no effect on each other. For example, it is assumed that the magnitude of initial investment doesn’t affect the operating costs.
For a project, the most expected case includes an initial investment of 150,000 dollars at the present time, an annual income of 40,000 for five years (starting from the first year), and a salvage value of 80,000. Evaluate the sensitivity of the project ROR to 20% and 40% increase and decrease in initial investment, annual income, project life, and salvage value.
Beforetax cash flow of this investment can be shown as:
$150,000

$40,000

$40,000  $40,000  $40,000  $40,000 
$80,000



0

1

2

3

4 
5

The most expected ROR based on the most expected initial investment, annual income, and salvage value can be calculated as:
$150,000=40,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)$
The most expected ROR will be 20.5%.
Initial investment  Change in prediction  ROR  Percentage change in 20.5% ROR prediction 

90,000  40%  43.5%  112.7% 
120,000  20%  29.6%  44.8% 
150,000  0  20.5%  0% 
180,000  20%  13.8%  32.6% 
210,000  40%  8.6%  57.8% 
As you can see, changes in ROR with respect to changes in initial investment are considerably high. In general, parameters that are close to time zero have a higher impact on the ROR of the project.
Project life  Change in prediction  ROR  Percentage change in 20.5% ROR Prediction 

3  40%  12.9%  36.6% 
4  20%  17.7%  13.5% 
5  0  20.5%  0% 
6  20%  22.2%  8.7% 
7  40%  23.4%  14.5% 
Note that changes in the project ROR become smaller as the project life gets longer.
Annual income  Change in prediction  ROR  Percentage change in 20.5% ROR prediction 

24,000  40%  8.1%  60.6% 
32,000  20%  14.3%  30.0% 
40,000  0  20.5%  0% 
48,000  20%  26.5%  29.5% 
56,000  40%  32.4%  58.5% 
Changes in annual income also have a significant effect on ROR because these changes start happening close to present time.
salvage value  Change in prediction  ROR  Percentage change in 20.5% ROR prediction 

48,000  40%  17.0%  17.0% 
64,000  20%  18.8%  8.2% 
80,000  0  20.5%  0% 
96,000  20%  22.0%  17.7% 
112,000  40%  23.5%  14.8% 
We can conclude that salvage value has the least effect on the ROR of the project because salvage value is the last amount in the future and its present value is relatively small compared to other amounts.
The following figure displays a tornado chart that is a very useful method to graphically summarize the results of sensitivity analysis. The right and left hand side of each bar indicate the maximum and the minimum ROR that each parameter generates when changed from 40% to +40%.
Type  Rate of Return Range 

Initial investment  8.6%  43.5% 
Project life  13%  23.4% 
Annual payment  8.1%  32.4% 
Salvage  17%  23.5% 
Please watch the following video (18:02): Sensitivity Analysis.
If you are interested, the following video (10:48) explains how to draw a tornado chart in Microsoft Excel (please watch from 6:10 to 9:00).
The expected value is defined as the difference between expected profits and expected costs. Expected profit is the probability of receiving a certain profit times the profit, and the expected cost is the probability that a certain cost will be incurred times the cost.
A wheel of fortune in a gambling casino has 54 different slots in which the wheel pointer can stop. Four of the 54 slots contain the number 9. For a 1 dollar bet on hitting a 9, if he or she succeeds, the gambler wins 10 dollars plus the return of the 1 dollar bet. What is the expected value of this gambling game? What is the meaning of the expected value result?
$$\begin{array}{c}\text{ProbabilityofSuccess}=4/54\\ \text{ProbabilityofFailure}=50/54\\ \text{ExpectedValue}=\text{ExpectedProfit}\text{ExpectedCost}=\left(4/54\right)*10\left(50/54\right)*1=\$0.185\end{array}$$ 0.185 dollars indicates that if the gambler plays this game over and over again, the average gain for the gambler per bet equals  0.185 dollars, which means the gambler will lose 0.185 dollars per bet. Note that for a satisfactory investment, a positive expected value is a necessary, but not sufficient, condition.
Assume drilling a well costs 400,000 dollars. There are three probable outcomes:
a) 70% probability that the drilled well is a dry hole
b) 25% probability that the drilled well is a producer well with such rate that can be sold immediately at 2,500,000 dollars
c) 5% probability that the drilled well is a producer well with such rate that can be sold immediately at 4,000,000 dollars
Calculate the project's expected value.
Note that +425,000 dollars is a statistical term; it means the average of +425,000 dollars will be achieved in the longterm for drilling over and over again in a repeated investment of this type.
Assume a research project that has the initial investment cost of 100,000 dollars. There are two possible outcomes:
a) 30 % success: that leads to an annual profit of 60,000 dollars for five years (starting from year 1) with a salvage value of zero
b) 70 % failure: that leads to annual profit and salvage value of zero
Considering a minimum 12% discount rate, compare the expected NPV, and explain if this investment is satisfactory.
30 % success:  $100,000  $60,000  $60,000  $60,000  $60,000  $60,000 
70 % failure:  $100,000  0  0  0  0  


0  1  2  3  4  5 
Since considering risk in calculations results in negative expected Net Present Value (ENPV), it can be concluded that this investment is expected to be economically unsatisfactory. Note that riskfree NPV (assuming 100% success probability) shows good and economically satisfactory results.
$$\text{RiskfreeNPV:}60,000\left(P/{A}_{12\%,5}\right)100,000=\$116,287$$Calculate the expected Rate of Return for the above example.
Expected ROR is the “i” that makes Expected NPV equal 0.
Expected Present worth income @ "i" – Present Worth Cost @"i" = 0
$\begin{array}{c}0.3\left(60,000\left(P/{A}_{i,5}\right)\right)=0.3*100,000+0.7*100,000\\ 0.3\left(60,000\left(P/{A}_{i,5}\right)\right)=100,000\end{array}$
By trial and error, Expected ROR =  3.4%
Note that risk free ROR shows a satisfactory result.
$60,000\left(P/{A}_{i,5}\right)=100,000$
Riskfree ROR = 52.8%, which is much higher than the minimum ROR.
Another way to calculate the expected ROR, which is similar to the previous method, is to calculate expected cash flow and then find the ROR for that.
Expected cash flow can be determined by multiplying each scenario’s cash flow by its probability and then make summation over each year:
Year 1  Year 2  Year 3  Year 4  Year 5  

Expected cash flow  $$\begin{array}{c}0.3*\left(\$100,000\right)\\ +0.7*\left(\$100,000\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$ 
Then:
Year 1  Year 2  Year 3  Year 4  Year 5  

Expected cash flow  $100,000  $18,000  $18,000  $18,000  $18,000 
By trial and error, Expected ROR =  3.4%
Please watch the following video (14:01): Expected Value Analysis, Part I.
Calculate Expected NPV for a minimum ROR 20% to evaluate the economic potential of buying and drilling an oil lease with the following estimated cost, revenues, and success probabilities.
The lease would cost 100,000 dollars at time zero and it is considered 100% certain that a well would be drilled to the point of completion one year later for a cost of 500,000 dollars. There is a 60% probability that well logs look good enough to complete the well at year 1 for a 400,000 dollar competition cost. If the well logs are unsatisfactory, an abandonment cost of 40,000 dollars will be incurred at year 1. If the well is completed, it is estimated there will be a 50% probability of generating production that will give 450,000 dollars per year net income for years 2 through 10 and a 35% probability of generating 300,000 dollars per year net income for years 2 through 10, with a 15% probability of the well completion being unsuccessful, due to water or unforeseen completion difficulties, giving a year 2 salvage value of 250,000 dollars for producing equipment.
The above decisionmaking process can be displayed in the following figure. These types of graphs are called decision trees and are very useful for risk involved decisions. Each circle indicates a chance or probability node, which is the point at which situations deviate from one another. (Costs are shown in thousands of dollars.)
Note: Times 1 and +1 are the same points in time and both indicate the end of year 1. The main body (of the tree) starts from the first node on the left with a time zero lease cost of 100,000 dollars that is common between all four situations. The next node, moving to the right, is the node that includes a common drilling cost of 500,000 dollars. At this node, an unsatisfactory and abandonment situation with a cost of 40,000 dollars in the first year (situation D) deviates from other situations (a branch for situation D deviates from the tree main body). The next node on the right (third node) is the node where situations A, B, and C (three separate branches) get separated from each other. At the beginning of each branch is the probability of that situation, and at the end of it, amounts due to that situation (including cost, income, and salvage value) are displayed.
So, there are four stations:
Situation A: Successful development that yields the income of 450 dollars per year
Situation B: Successful development that yields the income of 300 dollars per year
Situation C: Failure that yields a salvage value of 250 dollars at the end of year two
Situation D: Failure that yields abandonment cost of 40 dollars at the end of year one
Probability of situation A can be calculated as $P=0.5*0.6=0.3$
Probability of situation B can be calculated as $P=0.35*0.6=0.21$
Probability of situation C can be calculated as $P=0.15*0.6=0.09$
Probability of situation D can be calculated as $P=0.4$
Probability  
A) 
0.3

C=$100  C=$500+$400  I=$450  I=$450  ...  I=$450 
B) 
0.21

C=$100  C=$500+$400  I=$300  I=$300  ...  I=$300 
C) 
0.09

C=$100  C=$500+$400  Salvage=$250  0  ...  0 
D)  0.4  C=$100  C=$500+$40  0  0  ...  0 


0  1  2  3  ...  10 
Note that the summation of all properties should equal 1.
Project ENPV is the summation of ENPV for all situations. So, first, we need to calculate ENPV for each situation:
$$\begin{array}{c}A:\text{}\left(0.3\right)\left[100900\left(P/{F}_{20\%,1}\right)+450\left(P/{A}_{20\%,9}\right)\left(P/{F}_{20\%,1}\right)\right]\\ B:\text{}\left(0.21\right)\left[100900\left(P/{F}_{20\%,1}\right)+300\left(P/{A}_{20\%,9}\right)\left(P/{F}_{20\%,1}\right)\right]\\ C:\text{}\left(0.09\right)\left[100900\left(P/{F}_{20\%,1}\right)+250\left(P/{F}_{20\%,2}\right)\right]\\ D:\text{}\left(0.4\right)\left[100540\left(P/{F}_{20\%,1}\right)\right]\end{array}$$And it can be summarized in Table 61 as:
Probability  Year 1  Year 2  Year 3  Year 4  ...  Year 5  ENPV  

A  0.3  $100  $900  $450  $450  ...  $450  $198.5 
B  0.21  $100  $900  $300  $300  ...  $300  $33.1 
C  0.09  $100  $900  $250  0  ...  0  $60.9 
D  0.4  $100  $540  0  0  ...  0  $220 
Project ENPV is slightly less than zero compared to the total project cost of 1 million dollars, therefore, slightly unsatisfactory or breakeven economics are indicated.
Please watch the following video (13:32): Expected Value Analysis, Part 2.
Italicized sections are from Stermole, F.J., Stermole, J.M. (2014) Economic Evaluation and Investment Decision Methods, 14 edition. Lakewood, Colorado: Investment Evaluations Co.
One method used to analyze the uncertainty and risk involved in natural disaster decision makings is to choose the best alternative base on the lowest expected cost. In the following example, you can practice this method.
A company is planning to build a new plant. The plant requires water for its production process and needs to be built near a river. But the location has the probability of being flooded and building levees around the plant is necessary to protect the facility. There are four possible sizes of levee that have different costs, maintenance, and level of protection, as displayed in following table. Calculate the expected annualized cost for each levee, considering minimum ROR of 12% and 18 years project life. Then explain which levee has the lowest expected annualized cost for the company.
Levee size  Levee Cost  Probability that levee fails  Expected Damage  Annual maintenance 

1  $150,000  0.25  $100,000  $3,000 
2  $180,000  0.15  $130,000  $4,500 
3  $200,000  0.08  $140,000  $5,000 
4  $220,000  0.04  $180,000  $7,000 
Probability of levee failure: Probability of a flood exceeding levee size during the year
Expected Damage: Expected damage if flood exceeds levee size
In order to calculate expected annualized cost for each levee size, we need to convert all the costs into annual base. Then:
$$\text{Expectedannualcost=equivalentannualleveecost+expecteddamageperyear+annualmaintenance}$$From Table 112, equivalent annualized levee cost can be calculated as:
$$\left(Levee\text{}Cost\right)*\left(A/{P}_{12\%}{,}_{18}\right)=\left(Levee\text{}Cost\right)*0.13794$$
Expected damage per year is the multiplication of Probability of levee fails by Expected Damage
Expected annualized cost for different sizes of levee can be calculated as:
Levee size  Annual Levee Cost  Expected damage per year  Annual maintenance  Expected annual cost 

1  $20690.59  $25,000  $3000  $48,690.59 
2  $24828.72  $19,500  $4500  $48,828.72 
3  $27587.46  $11,200  $5000  $43,787.46 
4  $30346.21  $7,200  $7000  $44,546.21 
Results show that the third levee has the lowest expected annualized cost; therefore, it is the best alternative.
Sensitivity analysis is a means of identifying those critical variables that if changed, could considerably impact profitability measures such as rate of return or net present value. Risk analysis identifies the likelihood of project failure and the subsequent cost to the investor.
In this lesson, sensitivity analyses for NPV, ROR, project life, and annual payments are practiced. Expected NPV and ROR are also explained to help analyze the effects of risk and uncertainty on the project economics.
You have reached the end of Lesson 6! Doublecheck the todo list on the Lesson 6 Overview page [1] to make sure you have completed all of the activities listed there before you begin Lesson 7.