1. ESD.70J Engineering Economy
Fall 2007
Session Two
Michel-Alexandre Cardin –
Prof. Richard de Neufville –
ESD.70J Engineering Economy Module - Session 2

2. Session two – Simulation
Objectives:
Generate random numbers
Get familiar with Monte Carlo simulation
Set up simulation using Data Table
Generate statistics for simulation
Draw histogram and cumulative distribution function (CDF)
Also called Value At Risk and Gain (VARG) curve
ESD.70J Engineering Economy Module - Session 2

3. Questions for “Big vs. Small”
From the base case spreadsheet, we’ve calculated NPVs
However, we assumed deterministic demand forecasts for years 1, 2, and 3. This assumption is over-simplifying since actual demand will vary
 Since life in uncertain, we’d like to simulate a range of possible NPV outcomes, the Min, Max, distributions, and the expected NPV values!
ESD.70J Engineering Economy Module - Session 2

4. Set up random generator
Open ESD70session2-1.xls
/ESD70session2-1.xls
ESD.70J Engineering Economy Module - Session 2

5. Excel’s RAND() function
Returns random number greater than or equal to 0 and less than 1, sampled from a uniform distribution
To generate a random real number between a and b, use: =RAND()*(b-a)+a
In tab “RAND”, the formula in cell C3: “=Entries!C9*((1-Entries!C25)+2*Entries!C25*RAND())”
Returns a uniformly distributed random demand for year 1 centered around 300, which may differ by plus or minus 50%
Same logic applies for cell C4 and C5
Here a = (50% x 300) = 150; b = (50% x 300) = 450; p = 50% m = 300; a = m - pm = m(1 - p); b = m + pm = m (1 + p)
The formula RAND() x (b-a) + a means that you vary between 0% and 100% of the difference between a and b, and you add that random value to a. This creates a number between a and b, centered around the mid-value between a and b. E.g. if m = 300, a = 150 and b = 450. Then b - a = 300, and the RAND() can make vary the value RAND()(b-a) between 0 and 300 randomly. Therefore, a number varying between 0 and 300 added to a (or 150), will generate values between 150 and 450, or between a and b.
This formula expands as follows, which gives rise to the formula shown above:
RAND()(b-a) + a = RAND()(m + pm - m + pm) + m - pm = RAND()2pm + m - pm = m(2RAND()p p) = m(1 - p + 2pRAND()) = Entries!C9*((1-Entries!C25)+2*Entries!C25*RAND())
Because Entries!C9 = m = 300 and Entries!C25 = p = 50%
ESD.70J Engineering Economy Module - Session 2

6. Random number generator
Follow the instructions, step by step
Go to tab “RAND”
Type “=Entries!C9*((1-Entries!C25)+2*Entries!C25*RAND())” in cell C3
Type “=Entries!C10*((1-Entries!C25)+2*Entries!C25*RAND())” in cell D3
Type “=Entries!C11*((1-Entries!C25)+2*Entries!C25*RAND())” in cell E3
Press “F9” several times to see want happens
ESD.70J Engineering Economy Module - Session 2

7. Random number generator
Click “Chart” under “Insert” menu
“Chart Type” select “XY(Scatter)”, “Chart sub-type” select any one with lines, click “Next”
“Data Range” select B2:E3, click “Next”
“Chart options” select whatever pleases you, click “Next”
Choose “As object in” and click “Finish”
Press “F9” several times to see want happens
We have built a random demand generator for the 3 years that assumes independent demand (0 correlation) from year to year
ESD.70J Engineering Economy Module - Session 2

8. Check with your neighbors… Check the solution sheet… Ask me questions…
Give it a try!
Check with your neighbors…
Check the solution sheet…
Ask me questions…
ESD.70J Engineering Economy Module - Session 2

9. How Monte Carlo Simulation works
Calculate two NPVAs corresponding to the two random demand simulations
Demand in Year 1
Demand in Year 2
Demand in Year 3
NPVA
345
678
1001
189
579
690
How about generating many sets of random demands, and get the corresponding NPVAs automatically?
ESD.70J Engineering Economy Module - Session 2

10. Monte Carlo Simulation
ESD.70J Engineering Economy Module - Session 2

11. Setup simulation by Data Table
Follow these instructions, step by step:
Link demand in sheet for Plan A to the random demand generator, specifically, Plan A!E5 = Rand!C3; Plan A!G5 = Rand!D3; Plan A!I5 = Rand!E5
In “Simulation” sheet, type “=‘Plan A’!C16” in cell B8 (“=‘Plan A’!C16” is the output of result for NPVA)
Create the Data Table. Select “A8:B2008”, click “Table” under “Data” menu, in “column input cell” put “A7”, leave “row input cell” blank.
Same thing done for Plan B
NOTE: there is no input in the value column of the Data Table; an empty cell is selected as the “column input cell”. Why?
ESD.70J Engineering Economy Module - Session 2

12. ESD.70J Engineering Economy Module - Session 2
Explanation
For the One-Way Data Table, there is no need to set up the input values in a list, since each row of the Data Table calls RAND() and generates an NPVA projection
We have 2000 rows in the Data Table, so we have simulated 2000 times
Click “F9” to try another simulation run
ESD.70J Engineering Economy Module - Session 2

13. ESD.70J Engineering Economy Module - Session 2
Excel crashing note
If your Excel crashes during simulation runs, input some numbers (0’s or whatever) into the input value column to the left of the data series. Do not leave the area of input values blank in the Data Table
You can hide the dummy values by setting their font value to “white” color
ESD.70J Engineering Economy Module - Session 2

14. Check with your neighbors… Check the solution sheet… Ask me questions…
Give it a try!
Check with your neighbors…
Check the solution sheet…
Ask me questions…
ESD.70J Engineering Economy Module - Session 2

15. Calculating descriptive statistics
Useful to know mean, maximum, and minimum values for the simulated results
Follow step by step:
In Cell D1 type “=AVERAGE(B$9:B$2008)”
In Cell D2 type “=MAX(B$9:B$2008)”
In Cell D3 type “=MIN(B$9:B$2008)”
ESD.70J Engineering Economy Module - Session 2

16. Check with your neighbors… Check the solution sheet… Ask me questions…
Give it a try!
Check with your neighbors…
Check the solution sheet…
Ask me questions…
ESD.70J Engineering Economy Module - Session 2

17. Deterministic vs. dynamic results
From the base case spreadsheet, we learn NPVA is $162.1 million
What is your result for the expected NPVA and NPVB when considering demand uncertainty?
Jensen’s inequality and the Flaw of Averages:
Flaw of averages discussion…. Read the paper on The example here states that if an investor bases a retirement plan decision on the historical expected return of stocks, this strategy may run all of its funds much quicker than originally planned. This is because forecasts that represent or are based on averages are always wrong. In this case, a forecast based on average expectations of demand will inevitably lead to an NPV that is different than if one takes different demand scenarios, calculate NPV for each scenario, and take an expected value of NPV considering all possible scenarios. Jensen’s inequality represents this issue.
Jensen’s Inequalities study convex functions and the relationship between their mean values. They are stated as follows: If X is a real-valued random variable with E(|X|) finite and the function g() is convex, then E[g(X)] >= g(E[X]). The case is opposite for concave functions. Although this inequality is used in mathematics and statistics, it is also useful in economics when showing that an investor with a concave utility function prefers a certain return to the same expected return with uncertainty. For an investor with a convex utility function, the case is similar.
In this inequality, x can be demand for year 1 and f(x) be the corresponding NPV for that year. To be precise then, y could be demand for year 2, z demand for year 3, and therefore f(x, y, z) = NPV. Concentrating on one year for this example, on the left side of the inequality is the expected demand for year 1 E(x) so that f[E(x)] is the NPV corresponding to this expected demand value. This corresponds to the deterministic case since we assume that the demand in each year 1, 2, and 3 is the expected demand value, which produces the NPV of $162.1M. On the right side of the inequality is f(x), therefore the NPV obtained for year 1 for each possible value of demand. E[f(x)] is therefore the average of all NPV values obtained, where each NPV value is associated to a particular value of demand at year 1. This corresponds to what we do in simulations and is about $97M. The inequality states that the NPV value obtained from average or expected demand (deterministic case) is different from the NPV value obtained from the average of all NPV values obtained for each demand value (simulations).
Here, E[f(x)] = $97M is lower than f[E(x)] = $162.1M, so it seems f(x) is a concave function, and that the deterministic case is not representative of the real NPV values that may occur in reality. While investors would expect $162.1M, in reality and on average the project would produce much less value when one recognizes uncertainty.
ESD.70J Engineering Economy Module - Session 2

18. Value at Risk and Gain (VARG)
The VARG curve is another name for cumulative distribution function (CDF)
In our case, VARG curve aims at making a representation to managers that
“There is a probability X that NPV will be lower (higher) than Y dollars for this project”
Value At Risk is a common language on Wall Street. It stresses downside risk, though we should also look at CDF for upside potential of a project. Hence the name Value At Risk AND GAIN!
ESD.70J Engineering Economy Module - Session 2

19. Cumulative Distribution Function
Follow the instructions, step by step:
In sheet “Simulation”, set Cell G7 “=$D$3+($D$2-$D$3)/20*F7”, and drag the formula down to G27
Set Cell H7 “=COUNTIF($B$9:$B$2008,"<="&G7)”, and drag the formula down to H27
Set Cell I7 “=H7/2000”, and drag down to cell I27
Same is done for Plan B
Step 1 sets the first bounds as the MIN value, takes the range of values (as MAX - MIN) and divides it into an equal number of 20 bins. Then dragging the formula creates all 20 bins by multiplying with values between 0 and 20.
Step 2 counts the number of samples that are <= to the value of the bound.
Step 3 calculates the frequency of occurrence of samples for each bin.
ESD.70J Engineering Economy Module - Session 2

20. Cumulative Distribution Function
Right-click the chart on the right, select “Source Data”
Select “Series”, and press “Add”. This adds a new data series to the graph. Call it “NPVA”
Select the range =Simulation!$G$7:$G$27 for X values, and the range =Simulation!$I$7:$I$27 for Y values. Click “OK”
Double click the curve and change “Weight” to 3
Hit “F9” and watch the CDF move !
ESD.70J Engineering Economy Module - Session 2

21. ESD.70J Engineering Economy Module - Session 2
Explanation
We set up 20 data buckets and count how many data points fall into each interval
“=COUNTIF()” function counts the number of cells within a range that meet the criteria
The Excel file demonstrates how you can:
add NPVA and NPVB means as vertical lines
add histograms for two NPV distributions using the information created earlier
Can also use the Histogram analysis tool in “Data Analysis” package
ESD.70J Engineering Economy Module - Session 2

22. Cumulative Distribution Function
Use your cursor on the graph to find different Values At Risk and Gain
Alternatively, use the percentile function
In cell N5, type 10%
In cell R5, type
“=PERCENTILE(B9:B2008,N5)”
What does this tell you?
That’s interesting information for managers and decision-makers!
ESD.70J Engineering Economy Module - Session 2

23. ESD.70J Engineering Economy Module - Session 2
Question
Why are high NPV values more cut off for Plan B on the VARG curve and histogram than for Plan A?
A matter of constraints…
Plan A builds for capacity 900 right away in the first year. When demand is high in year 1 and 2, it can capture such extra demand, sell more, and therefore benefit from this unexpected opportunity. In year 3, demand higher than 900 cannot be captured, which is why the histogram for Plan A is more cut off on the right than on the left.
Plan B builds for capacity 300 in year 1, 600 for year 2, and 900 for year 3. Therefore in each year, when demand is higher than expected it cannot capture this unexpected opportunity and benefit from higher NPVs. This is why the VARG curve and histogram for Plan B have fewer high NPV values than Plan A. This also explains why the mean NPV for Plan B is lower than for Plan A.
ESD.70J Engineering Economy Module - Session 2

24. Check with your neighbors… Check the solution sheet… Ask me questions…
Give it a try!
Check with your neighbors…
Check the solution sheet…
Ask me questions…
ESD.70J Engineering Economy Module - Session 2

25. ESD.70J Engineering Economy Module - Session 2
Summary
Random number generation is fairly straight forward in Excel
At least two ways to run Monte Carlo simulation:
Direct RAND() calls
Using Data Table
Descriptive statistics from simulations
Mean, Max, Min, VARG curve
ESD.70J Engineering Economy Module - Session 2

26. ESD.70J Engineering Economy Module - Session 2
Next class…
Today’s session modeled demand uncertainty based on a uniformly distributed random variable
This is not a particularly realistic model, though it is simple and sufficient for today’s purposes
Next session we’ll explore alternative probability distributions from which to sample
ESD.70J Engineering Economy Module - Session 2