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CHAPTER 4 Selecting Crystal Ball Assumptions his chapter reviews basic concepts of probability and statistics using graphics from Crystal Ball’s distribution gallery, a portion of which is shown in Figure 4.1. If you have not had a class in basic probability and statistics at some point in your life or you need a refresher on these topics, consult a business statistics textbook such as Mann (2007). This chapter is intended to show the basics of how to specify probability distributions to be used as stochastic assumptions with Crystal Ball. Version 7.2 of Crystal Ball has 20 distributions from which to choose when defining assumptions. To see them, click the All button at the upper left of the distribution gallery. Six basic distributions are described here along with the binomial distribution. T CRYSTAL BALL’S BASIC DISTRIBUTIONS Yes-No Probabilists named the Bernoulli distribution in honor of the mathematician who showed analytically around 1700 the truth of the intuitive notion that when a fair coin is tossed repeatedly, it will come up heads about 50 percent of the time. It is perhaps the simplest of all probability distributions. The random variable Y has the Bernoulli distribution if it can take only one of two possible values, y = 0 or y = 1. The value y = 1 is called a ‘‘success,’’ and y = 0 is called a ‘‘failure’’ in probability parlance. In Crystal Ball, the Bernoulli distribution is known as the yes-no distribution. Crystal Ball calls y = 1 ‘‘yes’’ and y = 0 ‘‘no’’ because these terms often make sense in a modeling context. For example, Figure 4.2 shows Crystal Ball’s yes-no distribution for Pr(yes) = 0.5, where y represents the number of heads obtained in one toss of a fair coin. ‘‘Yes’’ means a head was tossed so y = 1, while ‘‘no’’ means a tail was tossed so y = 0. Now consider the type of situation that drew Bernoulli’s interest. The spreadsheet segment in Figure 4.3 shows a simple model to be used for finding the number of heads observed when tossing a fair coin five times. Each of the assumptions in cells B3:B7 are yes-no distributions with Pr(yes) = 0.5, so each assumption cell will 36 Selecting Crystal Ball Assumptions 37 FIGURE 4.1 The basic distributions listed in Crystal Ball’s distribution gallery. contain 1 on approximately 50 percent of the trials and 0 on the remaining trials. Each assumption cell’s value is generated independently of the other cells’ values. The forecast in cell B8 has the formula =SUM(B3:B7). Of course, we need not use simulation to model this situation because it is easy to determine the forecast distribution analytically. However, simulating a situation for which we know the analytical solution can be comforting. If we get results with simulation that are in accord with the analytical results, then we have some assurance that simulation will provide good approximate answers to questions regarding situations where analytical results are difficult or impossible to attain. For a simple example of finding an analytical result, consider the spreadsheet model FiveTosses.xls shown in Figure 4.4, which shows each of the 25 = 32 combinations of 0s and 1s that can occur on five tosses of a fair coin. Each combination is equally likely to occur. The number of heads in each combination is 38 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 4.2 Yes-no distribution to represent getting a head (y = 1) on one toss of a fair coin. FIGURE 4.3 Spreadsheet segment showing model for determining the distribution of five flips of a fair coin. Cells B3:B7 are yes-no(0.5) assumptions, and their sum in cell B8 is a Crystal Ball forecast. Selecting Crystal Ball Assumptions 39 FIGURE 4.4 Spreadsheet segment showing model for determining the distribution of five flips of a fair coin. Cells B3:B7 are yes-no(0.5) assumptions, and their sum in cell B8 is a Crystal Ball forecast. found by summing across the row for each combination. So to find the probability of each of the possible numbers of heads, we simply divide the frequency of occurrence of {0, 1, 2, 3, 4, 5} by 32, the total number of combinations to get the probabilities listed in cells C11:C16 in Figure 4.4. These are the probabilities associated with the binomial(0.5,5) distribution used below. Binomial While not included in Crystal Ball’s distribution gallery list of basic assumptions, the binomial distribution is so closely related to the yes-no distribution that it is included here and used later in the chapter. The binomial(p,n) is the distribution of 40 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 4.5 Binomial(0.5,5) distribution to represent the number of heads on five tosses of a fair coin. the sum of a fixed number, n, of Bernoulli trials that all have the same probability of success, p. Thus, the problem of determining the distribution of the number of heads in five tosses of a fair coin can be solved by using one Crystal Ball assumption—the binomial(0.5,5) assumption shown in Figure 4.5. Figure 4.6 depicts a model that gives the same results as that in Figure 4.3 by using Crystal Ball to simply generate the number of heads in five tosses from the distribution in Figure 4.5, and displaying the results in the forecast defined in cell B4 with the Excel formula =B3. The forecast distribution in Figure 4.6 looks almost identical to the forecast distribution in Figure 4.3, because the differences are due only to sampling error. Discrete Uniform The discrete uniform(L,H) distribution assigns equal probability to the set of integers between L and H, inclusive. For L = 1 and H = 6, it is the probability distribution representing the number of spots showing on the top face of a fair die rolled randomly. To illustrate the use of the discrete uniform, consider a problem with which Sir Isaac Newton dealt in the seventeenth century (Anděl 2001). The problem can be stated as follows: ■ ■ ■ Player A has 6 fair dice and wins if he rolls at least one ace (one spot showing on the top face of a die). Player B has 12 fair dice and wins if he rolls at least two aces. Player C has 18 fair dice and wins if he rolls at least three aces. Which player has the greatest chance of winning? Selecting Crystal Ball Assumptions 41 FIGURE 4.6 Simple model to represent the number of heads observed on five tosses of a fair coin. Cell B3 is a binomial(0.5,5) assumption. Cell B4 is a forecast cell with the formula =B3. Most seventeenth-century gamblers felt that because the ratio of rolls to aces (6:1) is the same for each player, the probability of winning should also be the same for each player. Newton’s analytical solution to this problem uses the Binomial distribution and is now considered trivial by probabilists. However, we will use simulation to find the approximate values of each player winning and compare the results to Newton’s analytical solution. Figure 4.7 shows a spreadsheet model of the situation. The Newton.xls model uses 6 discrete uniform assumptions in cells B5:B10, 12 discrete uniform assumptions in cells E5:E16, and 18 discrete uniform assumptions in cells H5:H22 to simulate the result of rolling each die. All 36 of these assumptions resemble the distribution shown in Figure 4.8 for cell B5, which is discrete uniform on the integers {1, 2, 3, 4, 5, 6}. Each of the 36 discrete uniform distributions generates observations independently of the others during the simulation runs. In the cell to the immediate right of each die’s result in Newton.xls is an Excel IF function that checks to see whether the result is a ace or not. For example, cell C5 contains the command =IF(B5=1,1,0), which puts a 1 in cell C5 if the assumption in B5 delivers a 1 during any iteration, and a 0 otherwise. This is an example of an indicator variable, which is a useful modeling concept that we use often throughout this book. Cells C11, F17, and I23 find the sums of the indicator variables in the cells directly above them, C5:C10, F5:F16, and I5:I22, respectively. The cells 42 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 4.7 Spreadsheet segment showing model for Newton’s dice problem. FIGURE 4.8 Discrete uniform distribution used for modeling the roll of one die for Newton’s dice problem. 43 Selecting Crystal Ball Assumptions TABLE 4.1 Table of values for Index in the Crystal Ball function =CB.GetForeStatFN(Range,Index) and the corresponding forecast statistic. See Chapter 2 for a definition of each statistic. Index 1 2 3 4 5 6 7 8 9 10 11 12 13 Statistic Trials Mean Median Mode Standard deviation Variance Skewness Kurtosis Coefficient of variability Minimum Maximum Range Mean standard error labeled A Wins? B Wins? and C Wins? are indicator variables to detect when the number of aces for A is greater than or equal to one, the number of aces for B is greater than or equal to two, and the number of aces for C is greater than or equal to three, respectively. These cells, C12, F18, and I24 are defined as Crystal Ball forecast cells. Finally, cells C13, F19, and I25 use the =CB.GetForeStatFN(Range,Index) command to find the mean of each forecast. The arguments for this command are Range, which is simply a reference to a Crystal Ball forecast cell, and Index, which is an integer between 1 and 13. Specify the integer for Index that corresponds to the desired forecast statistic listed in Table 4.1. For example, we use Index = 2 in Newton.xls because we want the means of the indicator variables in cells C13, F19, and I25. The resulting means after 10,000 runs are shown in cells C13, F19, and I25 to be 0.6628, 0.6205, and 0.5933. These values can be compared to the known probabilities obtained with the binomial distribution: 0.6651, 0.6187, and 0.5973, respectively. Thus, the solution to the problem is that Player A has the greatest chance of winning, followed by Player B, then by Player C. In a later chapter, we will see how to determine the precision of the estimates from simulation models. Uniform The uniform distribution is the simplest of all continuous probability distributions. It has only two parameters, the minimum and maximum values. It produces any continuous value between the minimum and maximum with equal likelihood. The 44 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 4.9 Continuous uniform distribution used for modeling a firm’s revenue, where the minimum possible value is $90, the maximum is $110, and all values in between are equally likely to occur. uniform distribution depicted in the dialog window shown in Figure 4.9 models a situation where a firm’s revenues range from $90 to $110, and all values in between are equally likely to occur. In dialog windows for its discrete distributions, Crystal Ball displays the possible values of the random variable on the horizontal axis and the associated probabilities on the vertical axis, as in Figure 4.8. For continuous distributions such as the uniform, Crystal Ball does not display values on the vertical axis because probability for continuous random variables is associated with intervals on the horizontal axis and not with single values. Because they represent probabilities for intervals rather than single numbers, the plots for continuous distributions are graphs of probability density functions, or simply PDFs. Use the uniform distribution when you know the minimum and the maximum values, but not a most likely value. The uniform distribution is completely specified by its two parameters, Minimum and Maximum. Because all values between Minimum and Maximum are equally likely to occur, its PDF has a uniform height over that range. The uniform is sometimes called the ‘‘distribution of maximum ignorance,’’ and should be replaced with a better estimate if one becomes available in later stages of the modeling process. However, there are some situations where the uniform may be the best distribution; for example, to model (1) where a leak might occur on a pipeline, or (2) time to failure of a component after a ‘‘burn-in’’ period, but before the required time to replace it. The spreadsheet segment in Uniform.xls shown in Figure 4.10 models the situation where a firm’s revenues follow the uniform(90,110) distribution and Selecting Crystal Ball Assumptions 45 FIGURE 4.10 Spreadsheet model for situation where a firm’s revenues are modeled as uniform(90,110), and where expenses are modeled as uniform(40,60). The resulting distribution of profit is triangular(30,50,70). expenses follow the uniform(40,60) distribution. The difference, profit, is defined as a forecast in cell B5, and a forecast chart for it has been copied and pasted onto the spreadsheet. It can be shown with the mathematical method of convolution (e.g., see Vose 2000) that the theoretical distribution of profit in this example is triangular(30,50,70), which is verified by the forecast chart in Figure 4.10. Triangular The triangular distribution is appropriate for use when you have little or no data available, but you know the minimum, maximum, and most likely values of a random variable. The triangular distribution is completely specified by its three parameters, Minimum, Likeliest, and Maximum. These three values are sufficient to determine the triangular shape shown in the icon. Of course, Minimum must be less than Maximum, and Likeliest must be in between (or equal to one of) these values. Figure 4.11 depicts a triangular(90,100,110) distribution. The spreadsheet segment in Triangular.xls shown in Figure 4.12 models the situation where a firm’s revenues follow the triangular(90,100,110) distribution and expenses follow the triangular(45,50,55) distribution. The difference, profit, is