MBA 7020 Business Analysis Foundations Simulation July 11, 2005

1. MBA 7020 Business Analysis FoundationsSimulation July 11, 2005

2. Agenda What is Simulation Spreadsheet Simulation Simulation with Crystal Ball

3. Developing and Using Models Processes for Analyzing Problem/Opportunities Situations Model Representation Deterministic Analysis Probabilistic Analysis Evaluation of Alternatives Decision Analysis Models Statistical, OR/MS, Financial Models Sensitivity Analyses (1-way, n-way) Goal Seeking and Optimization Using Probabilities (Decision Analyses) Using Probability Distributions (Simulation) Using Heuristics, Consensus Methods Goal Maximization, Internal Consistency

4. Evaluating Uncertainty Using Models Probability Analyses for answering “what-if” questions Model Outcome Controllable Intermediate Uncontrollable Simple Linear Regression (One -Way Probabilistic) Regression Analyses Model uncertain relationships using data Multiple Linear Regression (N -Way Probabilistic) Monte-Carlo Simulation Model uncertain variables using probability distributions

5. What is Simulation? • Modeling a real system to learn about its behavior • The model is a set of mathematical and logical relationships • You can vary conditions to test different scenarios

6. Advantages / Disadvantages • Advantages of Simulation • Inexpensive to evaluate decisions before implementation • Reveals critical components of the system • Excellent tool for selling the need for change • Disadvantages of Simulation • Results are sensitive to the accuracy of input data • Garbage in, Garbage out • Intelligent agents using secret rules • Investment in time and resources

7. Types of Simulations

8. Simulating with a Spreadsheet • Simulations can be performed with spreadsheets alone. However, add-in software packages can enhance the capabilities of Excel. • Two Excel add-in packages that will be used are Crystal Ball and @Risk. • These add-ins offer additional random distributions and easy commands to set up and run many more iterations than could be run in Excel. • In addition, they automatically gather statistical and graphical summaries of the results.

9. Agenda What is Simulation Spreadsheet Simulation Simulation with Crystal Ball

10. A Capital Budgeting Example: Adding A New Product LineSpreadsheet: Spreadsheet_Simulation.xls • Airbus Industry is considering adding a new jet airplane (model A3XX) to its product line. The following financial information is available: • Tax depreciation on the new equipment would be $10,000 per year over the 4 year expected product life. • Salvage value of the equipment at the end of the 4 years is estimated to be 0. • Airbus’ cost of capital is 10% and tax rate is 34%. Startup Costs $150,000 Sales Price $ 35,000 Fixed Costs (per year) $ 15,000 Variable Costs (per year) 75% of revenues

11. If demand is known, then a spreadsheet can be used to calculate the net present value (NPV). For example, assume that the demand for A3XXs is 10 units for each of the next 4 years: =C9*$B$3 =$B$4 =$B$5 =C10*$D$2 =$D$4*C14 =C10-SUM(C11:C13) =C14 – C15 =C16 + C13 =NPV($D$3,C17:F17)+B17 =-$B$2

12. The Model with Random Demand • It is unlikely that demand will be the same every year. A more realistic model would be one in which demand each year is a sequence of random variables. • This model of demand is appropriate when there is a constant base level of demand that is subject to random fluctuations from year to year. • Sampling Demand with a Spreadsheet: Assume initially that the demand in a year will be either 8, 9, 10, 11, or 12 units with each value being equally likely to occur. • This is an example of a discrete uniform distribution. • Now, use the formula =INT(8 + 5*RAND() ) to sample from a discrete uniform distribution on the integers 8, 9, 10, 11, 12 . • Multiple trials can be performed by pressing the recalculation key for the spreadsheet (e.g., F9).

13. =INT(8+5*RAND() ) Using this formula results in random demands Hitting the F9 key would result in a different sample of demands, and possibly a different NPV. The demands are random variables, therefore, the NPV is also a random variable.

14. Evaluating The Proposal • Two questions need to be answered about the NPV distribution: • What is the mean or expected value of the NPV? • What is the probability that the NPV assumes a negative value (making the proposal to add the A3XX less attractive)? • To answer these questions, a simulation model must be built. To run the simulation automatically and capture the resulting NPV in a separate spreadsheet, use the Data Table command.

15. Start with a blank worksheet by clicking on the Insert menu and select Worksheet Next, rename this blank worksheet 100 Iterations

16. Click the Edit menu and choose Fill – Series. In the resulting dialog, select Series in Columns and enter a stop value of 100. Click OK to fill series. Type the starting value (1) in cell A2 and hit Enter, then return to cell A2.

17. Now select the range A2:B101 and click Data – Table. Add column titles and the following formula to cell B2.

18. Excel will recalculate the values and store the resulting NPV in the adjacent cells in column B. In the resulting dialog, enter C1 for the column input cell and click OK. Note that since a random number generator is used in the formula, you may get different values than these.

19. Next, click on the Edit – Paste Special menu option and in the resulting dialog, choose Values. Now, to turn the formulas into actual values upon which we can focus, first select the range of cells B2:B101, then click on the Edit – Copy menu.

20. To get a summary of the 100 iterations, use Excel’s built-in data analysis tool. Click on Tools – Data Analysis. If you do not have this option, click on the Add-in option on the Tools menu and in the resulting dialog, click on Analysis ToolPak. After clicking OK, the Data Analysis dialog will open. Select the Descriptive Statistics option and click OK.

21. In the resulting dialog, choose the Input Range (or $C$2:$C$101) to include the 100 iterations. Now click on Output Range and enter the cell where the output will be placed. In addition, select Summary Statistics and click OK.

22. The resulting analysis gives the estimated mean NPV and standard deviation. Downside Risk and Upside Risk: To get a better idea about the range of possible NPVs that could occur, look at the minimum and maximum NPVs.

23. In the resulting dialog, set the input range (or $C$2:$C$101) and choose to save the results in a worksheet called NPV Distribution. Make sure to check Cumulative Percentage and Chart Output. Distribution of Outcomes: Now we ask the question: How likely will these extreme outcomes occur? To answer this, examine the shape of the distribution of the NPV by creating a histogram. Click on Tools – Data Analysis and choose Histogram.

24. In the resulting analysis, the Frequency (column B) indicates the number of trials that fell into the bins (categories) defined by column A. The cumulative % column indicates the cumulative percentage of observations that fall into each category or bin.

25. The histogram gives a visual representation of the distribution of NPVs. Note that it is somewhat bell shaped.

26. How Reliable is the Simulation? • Now the two questions about the distribution can be answered: • What is the mean or expected value of the NPV? • In this trial, the mean is $12,100. • What is the probability that the NPV assumes a negative value (making the proposal to add the A3XX less attractive)? • In this trial, the probability is >15%. • The next questions to ask are: • How much confidence do we have in the answers from the first trial? • Would we be more confident if we ran more trials?

27. =$E$4-1.96*$E$8/SQRT($E$16) =$E$4+1.96*$E$8/SQRT($E$16) How Reliable is the Simulation? • For a 95% confidence interval, the formula is: estimated mean + 1.96(standard deviation) • In this case, the standard deviation is the standard error (the standard deviation divided by the square root of the number of trials). • Based on this trial, the upper and lower confidence limits are: • So, we have 95% confidence that the true mean NPV is somewhere between $9,679 and $14,521.

28. Agenda What is Simulation Spreadsheet Simulation Simulation with Crystal Ball

29. Simulating with Spreadsheet Add-ins – Crystal Ball • Spreadsheet add-ins such as Crystal Ball and @Risk simplify the process of generating random variables and assembling the statistical results. • To illustrate, return to the capital budgeting example.

30. A Capital Budgeting Example: Adding A New Product LineSpreadsheet: CrystalBall_Simulation.xls • Airbus Industry is considering adding a new jet airplane (model A3XX) to its product line. The following financial information is available: • Tax depreciation on the new equipment would be $10,000 per year over the 4 year expected product life. • Salvage value of the equipment at the end of the 4 years is estimated to be 0. • Airbus’ cost of capital is 10% and tax rate is 34%. Startup Costs $150,000 Sales Price $ 35,000 Fixed Costs (per year) $ 15,000 Variable Costs (per year) 75% of revenues

31. If demand is known, then a spreadsheet can be used to calculate the net present value (NPV). For example, assume that the demand for A3XXs is 10 units for each of the next 4 years: =C9*$B$3 =$B$4 =$B$5 =C10*$D$2 =$D$4*C14 =C10-SUM(C11:C13) =C14 – C15 =C16 + C13 =NPV($D$3,C17:F17)+B17 =-$B$2

32. The Model with Random Demand • It is unlikely that demand will be the same every year. A more realistic model would be one in which demand each year is a sequence of random variables. • This model of demand is appropriate when there is a constant base level of demand that is subject to random fluctuations from year to year. • Sampling Demand with a Spreadsheet: Assume initially that the demand in a year will be either 8, 9, 10, 11, or 12 units with each value being equally likely to occur. • This is an example of a discrete uniform • distribution. • Enter the discrete distribution in a two-column • format for Crystal Ball to be able to use it.

33. After installing Crystal Ball, an additional toolbar will be displayed in Excel. Place your cursor in cell C9 and click on the Define Assumption button.

34. Click Ok to open the Custom Distribution dialog. Click on the Data button. Click Custom in the resulting dialog.

35. Enter the cell range in which the discrete distribution resides and click OK. The resulting distribution will be displayed: Click OK again to accept.

36. Repeat these steps for years 2-4 (cells D9:F9) or use Crystal Ball’s copy data and paste data icons. Clicking on this button will randomly change the demand and the NPV, since each is a random variable. To get Crystal Ball to draw a new random sample of demands, simply click on the Single Step icon.

37. Evaluating The Proposal • In order to answer the two questions about the NPV distribution: • What is the mean or expected value of the NPV? • What is the probability that the NPV assumes a negative value (making the proposal to add the A3XX less attractive)? • We need to run the simulation automatically a number of times and capture the resulting NPV. • To do this using Crystal Ball, first set up the base case model and enter the RNGs (Random Number Generators) in cells C9:F9 as was previously illustrated.

38. Next, click on B19 (the NPV cell) and then on the Define Forecast button.

39. Click on the Run Preferences icon to change the Maximum Number of Trials to 500 and click OK. After clicking on the Define Forecast icon, the following dialog will appear: Click on the Large forecast window size and When Stopped (faster) display option in this dialog. Click Set Default and then click OK.

40. To begin the simulation, click on the Start Simulation button. The following dialog will be displayed upon completion of the 500 iterations. Clicking OK will automatically produce a histogram.

41. To look at the statistics from the simulation, click on View menu on the histogram and click on Statistics. Each run of the simulation will produce different numbers so your results may not match those shown here.

42. Downside Risk and Upside Risk • Downside Risk and Upside Risk: To get an idea of the range of possible NPVs that could occur, look at the minimum and maximum values in the statistic results. • Distribution of Outcomes: In order to answer other questions about the distribution of NPVs, we need to look at the shape of the distribution. • The previous histogram (which was automatically produced) gives a graphical view of the distribution. The shape of the distribution is definitely bell-shaped.

43. 19.2 % of the observed NPV values were less than or equal to 0. Other information can be requested from Crystal Ball. For example, suppose you want to determine the exact probability that the NPV will be non-positive (< 0). Click on View menu on the histogram and click on Frequency Chart. In the Crystal Ball histogram window, enter 0 in the cell in the lower right corner and hit enter.

44. Click on View – Percentiles in the Crystal Ball window to display the percentiles of the NPV distribution.

45. We can have 95% confidence that the true mean will fall in an interval of + 1.96 standard deviations about the estimated mean. How Reliable is the Simulation? • Now that the questions concerning the mean of the distribution and the probability of negative values has been determined, the next questions to answer are: • How much confidence do we have in these answers? • Would we have more confidence if we ran more trials?

46. Other Distributions of Demand • Originally, we started with equal mean demands of 10 for each period (year). Then, we allowed for random variation in mean demand (between 8 and 12 units). : • Now, assume the mean demand will stay the same over the next four years, somewhere between 6 and 14 units a year, with all values being equally likely. • This scenario can be modeled as a continuous, uniform distribution between 6 and 14. • In addition, we can explore the impact of different demand distributions on the NPV. When the mean demand is relatively small, a distribution called the Poisson distribution is often a good fit.

47. Other Distributions of Demand • The Poisson distribution is a one-parameter distribution. Specifying the mean of this distribution completely determines it. • The Poisson distribution is a discrete distribution and the Poisson random variable can only take on non-negative integer values. • Using Crystal Ball’s Distribution Gallery, we can easily sample from a discrete Poisson distribution or from a continuous uniform distribution. • First, indicate in Crystal Ball that the cell D6 will have the uniform distribution and that cells C9:F9 will have a Poisson distribution with a mean value driven by the value in cell D6.

48. With your cursor on cell D6, click on the Define Assumptions icon and choose Uniform as the distribution. Click OK.

49. In the resulting dialog, specify the range of the distribution to be a minimum of 6 and a maximum of 14, then click OK.

50. To specify the Poisson distribution, first select cell C9 then click on the Define Assumption icon . In the resulting dialog, select Poisson and click OK.