1. ESD.70J Engineering Economy Module - Session 21 ESD.70J Engineering Economy Fall 2006 Session Two Alex Fadeev - afadeev@mit.edu Link for this PPT: http://ardent.mit.edu/real_options/ROcse_Excel_latest/ExcelSession2.pdf

2. ESD.70J Engineering Economy Module - Session 22 Session two – Simulation Objective: –Generate random numbers –Set up simulation by Data Table –Generate statistics for simulation –Draw histogram and cumulative distribution function (CDF)

3. ESD.70J Engineering Economy Module - Session 23 Questions for “Big vs. small” From the base case spreadsheet, we’ve calculated NPV’s. However, we assumed deterministic demand forecasts for years 1, 2 and 3. It is an over- simplifying assumption since actual demand will vary. We need to evaluate the range of NPV outcomes, their Min, Max, distributions and the expected NPV values!

4. ESD.70J Engineering Economy Module - Session 24 Outline Set up random number generator How does Monte Carlo simulation work Set up simulation by Data Table Get descriptive statistics from the simulation Draw cumulative distribution function (CDF)

5. ESD.70J Engineering Economy Module - Session 25 Excel’s RAND() f’n Returns an evenly distributed random number greater than or equal to 0 and less than 1 To generate a random real number between a and b, use: =RAND()*(b-a)+a Formula in cell C3: “=Entries!C9*((1- Entries!C25)+2*Entries!C25*RAND())” Returns a uniformly distributed random demand for year 1 around 300, but may differ by plus or minus 50%. [or just: “ =Entries!C9*(RAND()+(1-Entries!C25))” ] The Same logic for cell C4 and C5

6. ESD.70J Engineering Economy Module - Session 26 Random number generator Follow the instructions, step by step 1.Click “Worksheet” under “Insert” to add a new sheet, name it “Rand” 2.Type in “Year” in cell B2, “Random demand” in cell B3, type “1”, “2”, “3” in cell C2, D2, E2 respectively 3.Type “=Entries!C9*((1- Entries!C25)+2*Entries!C25*RAND())” in cell C3 4.Type “=Entries!C10*((1- Entries!C25)+2*Entries!C25*RAND())” in cell D3 5.Type “=Entries!C11*((1- Entries!C25)+2*Entries!C25*RAND())” in cell E3 Link for Excel: http://ardent.mit.edu/real_options/ROcse_Excel_latest/Session2-2.xls

7. ESD.70J Engineering Economy Module - Session 27 6.Click “Chart” under “Insert” menu 7.“Standard types” select “XY(Scatter)”, “Chart sub-type” select any one with lines, click “Next” 8.“Data range” select B2:E3, click Next 9.“Chart options” select whatever pleases you, click “Next” 10.Choose “As object in” and click “Finish” 11.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 Random number generator (cont)

8. ESD.70J Engineering Economy Module - Session 28 Check the solution sheet. Please ask questions now…

9. ESD.70J Engineering Economy Module - Session 29 How Monte Carlo Simulation works Calculate two NPV A ’s corresponding to the two random demand simulations Demand in Year 1 Demand in Year 2 Demand in Year 3 NPV A 3456781001 189579690 How about generating many sets of random demands, and get the corresponding NPV A ’s

10. ESD.70J Engineering Economy Module - Session 210 Monte Carlo Simulation (Cont)

11. ESD.70J Engineering Economy Module - Session 211 Set up simulation by Data Table Follow these instructions, step by step: 1.Link demand in sheet for Plan A to the random demand generated, specifically, Plan A!E5 = Rand!C3; Plan A!G5 = Rand!D3; Plan A!I5 = Rand!E5 2.Click “Worksheet” under “Insert” menu to add a new sheet, and name the new sheet “Simulation” 3.In “Simulation” sheet, Type “NPVA” in cell A1, type “=‘Plan A’!C16” in cell B1 (“=‘Plan A’!C16” is the output of result for NPV A ) 4.Select “A1:B2001”, click “Table” under “Data” menu, in “column input cell” put “A2002”, leave “row input cell” blank

12. ESD.70J Engineering Economy Module - Session 212 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 (for example A2:B2) calls rand() and generates an NPV A projection We have 2000 rows in the Data Table, so we have simulated 2000 times Click “F9” to try another simulation run

13. ESD.70J Engineering Economy Module - Session 213 Excel crashing note If you Excel crashes during simulation runs, input some numbers (0’s or whatever) into the dummy input 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.

14. ESD.70J Engineering Economy Module - Session 214 Check the solution sheet. Please ask questions now…

15. ESD.70J Engineering Economy Module - Session 215 Calculating descriptive statistics Useful to know mean, maximum, and minimum values for the simulated results Follow step by step: 1.In “Simulation” sheet, type “Mean” in cell D1, “Max” in cell D2, “Min” in Cell D3 2.Cell E1 type in “=AVERAGE(B$9:B$2008)”, Cell E2 type in “=MAX(B$9:B$2008)”, cell E3 type in “=MIN(B$9:B$2008)”

16. ESD.70J Engineering Economy Module - Session 216 Deterministic vs. dynamic results From the base case spreadsheet, we learn NPV A is $162.1 million What is your result for the expected NPV A when considering demand uncertainty? Jensen’s inequality:

17. ESD.70J Engineering Economy Module - Session 217 Cumulative Distribution Function Follow the instructions, step by step: 1.In sheet “Simulation”, type “Bound” in G6, “Count” in H6, and “CDF” in I6 2.Enter “0, 1, 2, …, 20” in cells F7 to F27 3.Set Cell G7 “=$D$3+($D$2-$D$3)/20*F7”, and drag the formula down to G27 4.Set Cell H7 “=COUNTIF($B$9:$B$2008,"<="&G7)”, and drag the formula down to H27 5.Set Cell I7 “=H7/2000”, and drag down to cell I27

18. ESD.70J Engineering Economy Module - Session 218 6.Click “Chart” under “Insert” menu (or click ) 7.“Standard types” select “XY(Scatter)”, “Chart sub- type” select anything with lines, click “Next” 8.“Data range” select =Simulation!$G$7:$L$27”, click 9.Go to tab and define the two series in which we are interested (bound vs. CDF). 10.Click or, exploring tertiary settings. 11.In the chart, double click Value (Y) axis to adjust min/max unit range, grid lines, number formatting. 12.Hit “F9” and watch the CDF move !

19. ESD.70J Engineering Economy Module - Session 219 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. Session2-2.xls file demonstrates how you can: –add NPA A and NPA B means as vertical lines. –add histograms for two NPV distributions. Challenge: set # of data buckets as an input variable and adjust the code to work for any #.

20. ESD.70J Engineering Economy Module - Session 220 Check the solution sheet. Please ask questions now…

21. ESD.70J Engineering Economy Module - Session 221 Summary Random number generation is fairly straight forward (though not always stable) in Excel At least two ways to run Monte Carlo simulation: –direct RAND() calls –Using Data Table

22. ESD.70J Engineering Economy Module - Session 222 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 random # distributions.