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CHAPTER 2 Analyzing Crystal Ball Forecasts n this chapter, using an example of accumulating funds for retirement, we see the graphical and numerical summaries of forecasts that Crystal Ball provides automatically. This chapter serves as a review of elementary statistical analysis, focused on the standard output built into Crystal Ball. I SIMULATING A 50 – 50 PORTFOLIO Let’s say that you want to start saving for your retirement. You are 30 years old and wish to retire at age 60. You plan to put away an inflation-adjusted $10,000 per year, and would like to know how much wealth you will have accumulated after 30 years. At this point, you consider only two types of assets: stocks and bonds. If you had perfect foresight, you would know exactly what returns each investment would bring over the next 30 years. With that information, you wouldn’t need Crystal Ball and could optimize your portfolio by investing in only those assets that you knew would go up. Of course, no one has perfect foresight, so what do you do? In this chapter, we’ll consider an oversimplified model for investing retirement funds and use it to illustrate how to analyze Crystal Ball forecasts. Accumulate.xls Overview. For this model we assume that returns on stocks and bonds during the next 30 years will resemble (in a statistical sense) the returns that have been observed during the years 1926 to 2004. You have heard that diversification is a good thing to do when investing your money, so you decide initially to split the money you want to set aside for retirement each year by putting one half into stocks and one half into bonds. We call this the 50–50 portfolio, and model it in the Excel file Accumulate.xls. The model draws returns on stocks and bonds randomly for each year and calculates the wealth you will have accumulated for retirement 30 years from now, assuming a constant 3 percent inflation rate. A segment of this model is shown in Figure 2.1. Note that rows 14 through 40 are hidden to save space. 11 12 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 2.1 Spreadsheet segment from Accumulate.xls showing accumulated value after 30 years of retirement savings where 50 percent was put into stocks and 50 percent into bonds each year. Note that Rows 14 through 40 are hidden. Forecast. You want to see the possible distribution of wealth you will have accumulated after 30 years—Year 30 Wealth. Cell D4 has the formula =H41 to get Year 30 Wealth displayed near the top of the spreadsheet. Stochastic assumptions. Cells B12:B41 represent the total annual return on stocks for years 1 through 30. Each year’s return on stocks is lognormal with mean 1.1517 and standard deviation 0.2746. Cells C12:C41 represent the total annual return on bonds for years 1 through 30. Each year’s return on bonds is lognormal with mean 1.0585 and standard deviation 0.0735. In this model, all assumptions are statistically independent of each other (that is, all correlations are zero). The parameters of the lognormal distributions were chosen with the help of Crystal Ball’s distribution-fitting feature, which is described in Chapter 4. Decision variable. The sole decision variable in this model is Cell B4, the proportion of each year’s investment that is allocated to stocks. The proportion allocated to bonds is calculated in Cell B5 as =1–B4 to ensure that the entire investment is put into either stocks or bonds each year. The initial proportion allocated to stocks is 50 percent. Summary. Figure 2.2 shows the frequency view of a forecast chart for Year 30 Wealth obtained after running the simulation for 10,000 trials. Note that the distribution is highly skewed to the right because of a few instances of very high returns over the 30-year period. Frequency Chart The default graph in the Forecast window is a frequency chart (also known as a histogram) that shows how often a forecast cell had a value falling in each of several possible intervals. In the forecast window for Year 30 Wealth in Figure 2.2, the possible values range from $0 to $15 million, and are broken up into 15 equal Analyzing Crystal Ball Forecasts 13 FIGURE 2.2 Frequency chart showing accumulated value after 30 years of retirement savings where 50 percent was put into stocks and 50 percent into bonds each year. The estimated probability of accumulating between $1 million and $2 million is shown in the Certainty field. intervals (of size $1 million). The height of each bar indicates how many simulation trials resulted in a Year 30 Wealth value that fell in the corresponding interval. For example, the tallest bar in Figure 2.2 indicates that Year 30 Wealth was between $1 million and $2 million in just fewer than 3,600 trials. The right side of the histogram has the frequency scale. Because there were 10,000 trials run, and the estimated probability (certainty) of Year 30 Wealth being between $1 million and $2 million is 34.47 percent, we know that there must have been 3,447 trials that resulted in Year 30 Wealth between $1 million and $2 million. The left side of the histogram has the probability scale. The text ‘‘9,761 Displayed’’ in the upper right corner of the window indicates that Year 30 Wealth was greater than $15 million (literally, ‘‘off the chart’’) on 239 of the 10,000 simulation trials run. We know that all of the undisplayed values must be above $15 million rather than below $0 because the portfolio value can never be negative under the assumptions we used. Cumulative Frequency Chart Change the forecast window by clicking on Preferences, Forecast. . ., then selecting Cumulative Frequency in the View field and clicking OK. You may also click on View → Cumulative Frequency in the top menu of the forecast window. The cumulative frequency chart in Figure 2.3 shows the frequency with which the simulation trials fell into each interval or below. For example, from the cumulative frequency 14 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 2.3 Cumulative frequency chart showing accumulated value after thirty years of retirement savings where 50% was put into stocks and 50% into bonds each year. The estimated probability of accumulating more than $1 million is shown in the Certainty field. chart you can see that in 80.09 percent of the trials, 30-year wealth had a value below $5 million. Another way to interpret this example is to say that the estimated probability is approximately 80 percent that 30-year wealth will be less than $5 million. Note that Crystal Ball also provides a reverse cumulative frequency chart (not pictured here) that shows the frequency with which the simulation trials fell into each interval or above. You can cycle through the different views by using the keystroke combination Ctrl-d when the forecast window is active. Table 2.1 lists the keystroke combinations (sometimes called ‘‘hot keys’’ or ‘‘keyboard shortcuts’’) that are available to alter your view of the forecast window. Make the Year 30 Wealth forecast window active and experiment with different combinations to see the effects. Statistics View The statistics view of the Forecast window in Figure 2.4 provides numerical summaries of the forecast. Descriptions of the statistics computed automatically by Crystal Ball in Figure 2.4 are listed below. The number of each statistic in the list can be used in the Excel formula =CB.GetForeStatFn(Range,Index) as the value of Index. The statistics and their indices are also listed in Table 4.1, and usage of the =CB.GetForeStatFn(Range,Index) is illustrated in Cells D46:E57 in Accumulate.xls. 15 Analyzing Crystal Ball Forecasts TABLE 2.1 Keystroke combinations (‘‘hot keys’’) that can be used to cycle through settings available in the Chart Preferences dialog. These commands work on the primary distribution—the theoretical probability distribution for assumptions, and the generated values for forecasts and overlay charts. Hot Key Description Ctrl-d Cycles through chart views: Frequency, Cumulative Frequency, Reverse Cumulative Frequency (for assumption and forecast charts) Cycles through bins or group interval values to adjust the number of data bins used to create the chart Cycles through gridline settings: None, Horizontal, Vertical, Both Cycles through chart types: Area, Line, Column; for sensitivity charts: Bar (direction), Bar (magnitude), Pie (in Contribution to Variance view) Cycles between two-dimensional and three-dimensional chart display Cycles through central tendency marker lines: None, Base Case, Mean, Median, Mode (except for sensitivity and trend charts) Toggles the legend display on and off Cycles through percentile markers: None, 10%, 20%, . . ., 90% Cycles through window views when Excel is not in Edit mode: Chart, Statistics, Percentiles, Goodness of Fit (if distribution fitting is selected—except for trend charts) Ctrl-b or Ctrl-g Ctrl-l Ctrl-t Ctrl-3 Ctrl-m Ctrl-n Ctrl-p Spacebar For the mathematical expressions in the list below, we denote the values produced by the simulation in a forecast cell as y1 , y2 , . . . , yn , where n is the number of iterations run before the simulation stops. 1. Trials. The first item listed in the statistics view is Trials, which is also called the number of iterations, n. A trial (or iteration) is a three-step process in which Crystal Ball generates a random number for each assumption cell, recalculates the spreadsheet model(s), and collects the result(s) for the forecast window(s). The number of trials is the only descriptive statistic value that is under your direct control. Use the Run Preferences dialog box to specify the maximum number of trials. 2. Mean. The next item in the Statistics View is Mean, which is the same as the arithmetic average. It is calculated as 1
yi . Mean = Y = n i=1 n Even though the window indicates that only 9,761 trials are displayed, the value $3,775,824 for the mean in Figure 2.4 is calculated from all 10,000 trials of the simulation. 16 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 2.4 Forecast chart statistics for accumulated value after 30 years of retirement savings where 50 percent was put into stocks and 50 percent into bonds each year. If you wish to calculate statistics for selected values of the forecast, click on Preferences, Forecast..., then the Filter tab. You will be able to include or exclude any specified range of values from the calculations. This feature will come in handy later for calculating expected tail loss. The mean is one of three measures of location computed by Crystal Ball. The other two measures of location are the median and the mode. In a highly skewed distribution such as Year 30 Wealth, the mean may not be the best indication of the location of the distribution. Use the certainty range in the forecast chart to see for yourself that the probability of Year 30 Wealth equalling or exceeding the mean is only about 30 percent. 3. Median. The median is the value in the middle of the distribution. For example, 6 is the median of the distribution of values 1, 3, 6, 7, and 9. In a distribution with an odd number of values, the median is found by ordering the values from smallest to largest and then selecting the middle value. In a distribution with an even number of values, the median is equal to the mean of the two middle ordered values. For Year 30 Wealth, the median value in Figure 2.4 is $2,425,951, which means that the probability is about 50 percent that our retirement portfolio will be that large or larger at that time. The median is less sensitive to outliers than the mean. For that reason, it is sometimes preferred to the mean as a measure of location, as it is here because the Year 30 Wealth forecast frequency distribution is highly skewed. 17 Analyzing Crystal Ball Forecasts 4. Mode. The mode is the single value that occurs most frequently in a set of values. For a forecast that can take on continuous values, it is likely that no single value will occur more than once, so the mode is often listed as dashes (– –) in the statistics view, as it is in Figure 2.4. 5. Standard Deviation. The standard deviation is a measure of dispersion, or spread, of a distribution. Think of it as roughly equal to the average distance of each value from the mean, although as you can see in the formula below, it is not exactly equal to that:   n  1
(2.1) Standard Deviation = s =  (yi − y)2 . n − 1 i=1 The standard deviation is one of two equivalent measures of spread computed by Crystal Ball. The other measure of spread is the variance. In many applications, the standard deviation is preferred because it is measured in the same units as the forecast variable. 6. Variance. The variance is another measure of dispersion that is equivalent to the standard deviation. Because the variance is equal to the standard deviation squared, it sometimes appears in the statistics view as a very large number. The variance is calculated as: 1
(yi − y)2 . n − 1 i=1 n Variance = s2 = 7. Skewness. Skewness is a measure of asymmetry of a frequency distribution. The distributions pictured in Figure 2.5 display negative, positive, and near-zero skewness. The formula for skewness used by Crystal Ball is Skewness = 3 n  1
yi − y . n i=1 s The large positive skewness of Year 30 Wealth can be seen in Figure 2.2, and is measured as 32.20 in Figure 2.4. The fact that the mean of $3.776 million is so much larger than the median of $2.426 million is also evidence of large positive skewness. Alternative measures of skewness are 3(Mean − Median) 3(Mean − Mode) and , Standard Deviation Standard Deviation either of which you can calculate easily from the Crystal Ball output. With all else equal, positive skewness in accumulated wealth is desirable; however, note that here the large positive skewness makes it misleading to expect to earn the mean wealth. As calculated above, there is less than a 30 percent chance that you will actually accumulate the mean wealth or more. 18 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 2.5 Frequency distributions depicting negative (Skewness = −2), positive (+2), and near-zero (0.02) skewness at top left, top right, and bottom center, respectively. 8. Kurtosis. Kurtosis is a measure of peakedness, which is equivalent to measuring tail thickness. The formula for kurtosis used by Crystal Ball is 4 n  1
yi − y . Kurtosis = n i=1 S If two distributions have the same standard deviation, the one with the higher kurtosis will have a higher peak and heavier tails. All normal distributions have a kurtosis of 3.0, and frequency distributions with a kurtosis near 3.0 are called mesokurtic. Distributions that have significantly higher kurtosis are called leptokurtic and those with significantly lower kurtosis are called platykurtic. Some computer programs and authors subtract 3.0 from the definition above used by Crystal Ball. The technical name for this is then excess kurtosis, but you may see it too called kurtosis by some authors, so beware. In fact, the statistic calculated by Excel’s =KURT(ref) command and its Data Analysis tool is actually excess kurtosis—calculated by a slightly different formula than Crystal Ball—but is labelled simply as kurtosis. Figure 2.6 shows an overlay chart for two distributions that have mean zero and standard deviation one. The distributions with the lower peak is the standard normal distribution, which has kurtosis = 3.0, as does every normal distribution. The distribution labeled Mixture has kurtosis = 12.1. It could have come from a market in which 5% of the time returns have high variability and 95 percent of the time they have low variability (see the file Kurtosis.xls for details). 19 Analyzing Crystal Ball Forecasts FIGURE 2.6 Overlay chart depicting a mesokurtic standard normal distribution (kurtosis = 3) and a leptokurtic mixture distribution (kurtosis = 11.7). See the file Kurtosis.xls for details. 9. Coefficient of Variability. The coefficient of variability, also known as the coefficient of variation, is a relative measure of dispersion found as Coefficient of Variability = s . y It might be more useful than standard deviation for some purposes. For example, a standard deviation of 10 may be insignificant if the mean is 10,000 (giving a coefficient of variability = .001) but may be substantial if the mean is 100 (coefficient of variability = .1). Because the mean y and standard deviation s have the same units, the coefficient of variability is dimensionless. Crystal Ball calculates it routinely as part of the standard output, but the coefficient of variability is best used only when all simulated values are positive. When the values can be both positive or negative, the mean can be zero or negative. When the mean value is near zero, the coefficient of variability is sensitive to small changes in the standard deviation, which can limit its usefulness. When the mean value is negative, use the absolute value of the coefficient of variability to get a more meaningful relative measure of dispersion. 10. Minimum. The minimum is the smallest value of all the observed forecast values. Note for models using unbounded-on-the-left stochastic assumptions such as the normal distribution, the more trials that are run, the smaller the minimum is likely to be simply because there are more opportunities for Crystal Ball to generate extreme observations. 20 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL √ FIGURE 2.7 Plot of 100/ n for n in the interval [100, 10000]. This plot shows how the standard error of the mean decreases as a function of the number of trials in the simulation. Much of the decrease in standard error is gained after only 2,000 trials. 11. Maximum. The maximum is the largest value of all the observed forecast values. For assumptions that are unbounded on the right such as the normal or lognormal distributions, the more trials that are run, the larger the maximum is likely to be simply because there are more opportunities for Crystal Ball to generate extreme observations. 12. The Range is the difference between the minimum and the maximum. In versions of Crystal Ball before 7.2, the Range was listed in the statistics view of the forecast window. For backward compatibility, the command =CB.GetForeStatFN(range,12) will return the Range statistic, which is simply the Minimum subtracted from the Maximum. Do not confuse the CB.GetForeStatFN argument range, which represents the address of a forecast cell, with the Range statistic described here. 13. Mean Standard Error. The mean standard error is a measure of precision of the estimate of the mean. The smaller the mean standard error, the greater the precision. Figure 2.7 shows how the mean standard error decreases in a nonlinear manner as the number of trials increases for a forecast distribution having a standard deviation of 100. Most of the precision in the estimate is gained by 2,000 trials. Forecast Window Percentiles View The percentiles view in Figure 2.8 gives the forecast values that are just larger than the corresponding percent of all the values. For example, the 80th percentile for Year 30 Wealth is $4,985,234. This means that 80 percent of the forecast values were less than that amount, while 20 percent were greater than that amount. Note