1. Simulation II

2. Overview Advanced Simulation Applications Retirement Planning
Securities Pricing
Project Management
Operations -- Prof. Juran

3. Retirement Planning
Amanda has 30 years to save for her retirement. At the beginning of each year, she puts $5000 into her retirement account. At any point in time, all of Amanda's retirement funds are tied up in the stock market. Suppose the annual return on stocks follows a normal distribution with mean 12% and standard deviation 25%. What is the probability that at the end of 30 years, Amanda will have reached her goal of having $1,000,000 for retirement? Assume that if Amanda reaches her goal before 30 years, she will stop investing.
Operations Prof. Juran

4. Operations -- Prof. Juran

5. The range C6:C35 will be random numbers, generated by @Risk.
The annual investment activities (columns A-D, beginning in row 5) actually extend down to row 35, to include 30 years of simulated returns.
The range C6:C35 will be random numbers, generated
We could track Amanda’s simulated investment performance either with cell F5 (simply =D35, the final amount in Amanda’s retirement account), or with F4 (the maximum amount over 30 years). Using F4 allows us to assume that she would stop investing if she ever reached $1,000,000 at any time during the 30 years, which is the assumption given in the problem statement.
Cell H1 is either 1 (she made it to $1 million) or 0 (she didn’t). Over many trials, the average of this cell will be out estimate of the probability that Amanda does accumulate $1 million. This will be output cell.
Operations Prof. Juran

6. We create a graph showing the amount of money in Amanda’s retirement account during the simulation. This adds little to our understanding, but it’s fun to watch.
Operations Prof. Juran

7. Operations -- Prof. Juran

8. Operations -- Prof. Juran

9. Operations -- Prof. Juran

10. Operations -- Prof. Juran

11. It looks like Amanda has about a 48% chance of meeting her goal of $1 million in 30 years.
Operations Prof. Juran

12. Example: Asian Option
George Brickfield’s business is highly exposed to volatility in the cost of electricity.
He has asked his investment banker, Lisa Siegel, to propose an option whereby he can hedge himself against changes in the cost of a kilowatt hour of electricity over the next twelve months.
Operations Prof. Juran
Decision Models Prof. Juran

13. Lisa thinks that an Asian option would work nicely for George’s situation.
An Asian option is based on the average price of a kilowatt hour (or other underlying commodity) over a specified time period.
Operations Prof. Juran
Decision Models Prof. Juran

14. In this case, Lisa wants to offer George a one year Asian option with a target price of $0.059.
If the average price per kilowatt hour over the next twelve months is greater than this target price, then Lisa will pay George the difference.
If the average price per kilowatt hour over the next twelve months is less than this target price, then George loses the price he paid for the option (but he is happy, because he ends up buying relatively cheap electricity).
Operations Prof. Juran
Decision Models Prof. Juran

15. What is a fair price for Lisa to charge for 1 million kwh worth of these options?
Use the historical data provided and Monte Carlo simulation to arrive at a fair price.
Operations Prof. Juran
Decision Models Prof. Juran

16. Operations -- Prof. Juran
Decision Models Prof. Juran

17. Operations -- Prof. Juran
Decision Models Prof. Juran

18. Operations -- Prof. Juran
Decision Models Prof. Juran

19. Operations -- Prof. Juran
Decision Models Prof. Juran

20. Operations -- Prof. Juran
Decision Models Prof. Juran

21. E11 calculates the payout on the option (an @Risk output cell).
In B8:B19 we have distribution cells, normally distributed with the mean and standard deviation from our sample data (C3 and C4).
In C8:C19 we use the random percent returns to calculate monthly prices, which are averaged in E8 for the whole year.
E11 calculates the payout on the option output cell).
The average value of E11 over many trials will be a reasonable estimate of the fair price for this option.
Operations Prof. Juran
Decision Models Prof. Juran

22. Operations -- Prof. Juran
Decision Models Prof. Juran

23. Operations -- Prof. Juran
Decision Models Prof. Juran

24. The frequency chart indicates that the option is frequently worthless (as evidenced by the tall bar at zero), but that the payout is occasionally $20,000 or more.
To estimate a fair price, the most useful piece of the simulation output is the sample mean of approximately $2, per million kwh.
Operations Prof. Juran
Decision Models Prof. Juran

25. Operations -- Prof. Juran
Decision Models Prof. Juran

26. Operations -- Prof. Juran
Decision Models Prof. Juran

27. Beta Distribution
The Beta distribution is a continuous probability distribution defined by four parameters:
Parameter
Description
Characteristics
Min
Minimum Value
Any number -
∞ to ∞
Max
Maximum Value
Alpha ( α )
Shape Factor
Must be > 0
Beta ( β
Operations Prof. Juran

28. Operations -- Prof. Juran

29. The Beta distribution is popular among simulation modelers because it can take on a wide variety of shapes, as shown in the graphs above.
The Beta can look similar to almost any of the important continuous distributions, including Triangular, Uniform, Exponential, Normal, Lognormal, and Gamma.
For this reason, the Beta distribution is used extensively in PERT, CPM and other project planning/control systems to describe the time to completion of a task.
Operations Prof. Juran

30. Operations -- Prof. Juran

31. PERT Approximations
The project management community has evolved approximations for the Beta distribution which allow it to be handled with three parameters, rather than four.
The three parameters are the minimum, mode, and maximum activity times (usually referred to as the optimistic, most-likely, and pessimistic activity times).
This doesn’t give exactly the same results as the mathematically-correct version, but has important practical advantages.
Most real-life managers are not comfortable talking about things like probability functions and Greek-letter parameters, but they are comfortable talking in terms of optimistic, most-likely, and pessimistic.
Operations Prof. Juran

32. 3-step Procedure
Operations Prof. Juran

33. Operations Example: Project Management (PERT)
Sharon Katz is project manager in charge of laying the foundation for the new Brook Museum of Art in New Haven, Connecticut.
Liya Brook, the benefactor and namesake of the museum, wants to have the work done within 41 weeks, but Sharon wants to quote a completion time that she is 90% confident of achieving.
The contract specifies a penalty of $10,000 per week for each week the completion of the project extends beyond week 43.
Operations Prof. Juran

34. Operations -- Prof. Juran

35. What is the probability of completion by week 43?
What completion time should Sharon use, if she wants to be 90% confident?
What is the probability of completion by week 43?
Give an estimated probability distribution for the amount of penalties Sharon will have to pay.
What is the expected value of the penalty?
Which activities are most likely to be on the critical path?
Operations Prof. Juran

36. An activity-on-arc diagram of the problem:
Operations Prof. Juran

37. Model Overview
Operations Prof. Juran

38. Here’s the section keeping track of the activity times
Here’s the section keeping track of the activity times. The numbers in column I will distribution cells.
Operations Prof. Juran

39. @Risk
For each of the random activities, we create a distribution cell, as shown here for Activity A:
Operations Prof. Juran

40. Operations -- Prof. Juran

41. Example: Activity E
Operations Prof. Juran

42. Here’s the model after doing this for every random activity time (Activities D, F, I, L, M, and the Dummy activity have no variability):
Operations Prof. Juran

43. Now we set up an area in the spreadsheet to track each of the paths through the network, to see which one is critical. This network happens to have six paths, so we set up a cell to add up all of the activity times for each of these paths:
Operations Prof. Juran

44. Now, for each activity, we can set up a cell to indicate whether the activity was critical for any particular realization of the model.
Note that Activity H (Procure Steel, in row 9) is part of two paths (A-B-D-E-F-H-K-L-M, in row 19, and A-C-E-F-H-K-L-M, in row 20). In this example, neither of those was the critical path, so Activity H is non-critical.
Operations Prof. Juran

45. Here’s a cell to tell whether the project was completed by week 43:
Here’s a cell to keep track of the penalty (if any) Sharon will have to pay. Note that we have assumed that the penalty applies continuously to any part of a week.
Operations Prof. Juran

46. Now we create output cells to track the completion time of the whole project (B30) as well as the criticalities of the various paths (H19:H24) and activities (L2:L15).
We also make output cells to track whether the project took longer than 43 weeks, and what the penalty was.
Operations Prof. Juran

47. What completion time should Sharon use, if she wants to be 90% confident?
Operations Prof. Juran

48. What is the probability of completion by week 43?
Operations Prof. Juran

49. What is the probability of completion by week 43?
Operations Prof. Juran

50. What is the expected value of the penalty?
Give an estimated probability distribution for the amount of penalties Sharon will have to pay.
What is the expected value of the penalty?
Operations Prof. Juran

51. Which activities are most likely to be on the critical path?
Operations Prof. Juran

52. Summary Advanced Simulation Applications Retirement Planning
Securities Pricing
Project Management
Tonight?
Distribution fitting
Correlated distributions
Value-at-Risk (VaR)
Operations Prof. Juran