Simulation II

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

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

Operations -- Prof. Juran

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 by @Risk. 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 an @Risk output cell. Operations -- Prof. Juran

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

Operations -- Prof. Juran

Operations -- Prof. Juran

Operations -- Prof. Juran

Operations -- Prof. Juran

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

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

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

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

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

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

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

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

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

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

E11 calculates the payout on the option (an @Risk output cell). In B8:B19 we have 12 @Risk 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 (an @Risk 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

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

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

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 $2,934.46 per million kwh. Operations -- Prof. Juran Decision Models -- Prof. Juran

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

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

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

Operations -- Prof. Juran

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

Operations -- Prof. Juran

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

3-step Procedure Operations -- Prof. Juran

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

Operations -- Prof. Juran

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

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

Model Overview Operations -- Prof. Juran

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 be @Risk distribution cells. Operations -- Prof. Juran

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

Operations -- Prof. Juran

Example: Activity E Operations -- Prof. Juran

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

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

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

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

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

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

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

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

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

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

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