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Contents

The National Association of Certified Valuators and Analysts (NACVA) supports the users of business and intangible asset valuation services and financial forensic services, including damages determinations of all kinds and fraud detection and prevention, by training and certifying financial professionals in these disciplines. NACVA training includes Continuing Professional Education (CPE) credit and is available to both members and non-members. Contact NACVA at (801) 486-0600 or visit the web site at .

Title Page

To my children, Joshua David and Rachel Leah, who have exceeded my hopes and expectations.

—MGF

To my wife Lori, my sons, Daniel, James, and John. I thank them for their encouragement, love, support, and patience during this journey.

—JAD

Preface

From years of presenting at conferences and seminars, participating in roundtable discussions and case study analyses, and mentoring fellow practitioners, it became obvious to us that the typical ABV, CBA, ASA, or CVA who was attempting to calculate economic damages either as a stream of lost profits or as lost value of the business using the direct market data method had little knowledge of either statistical methods or the advantages to be obtained by applying them to the task at hand.

This book is intended for practitioners who have some experience in the field of calculating economic damages and who are looking to acquire some new tools for their toolkit—tools that are more sophisticated and flexible than simple averaging techniques. These typical practitioners will remember little from their college statistics course and will not have access to or be capable of using stand-alone statistical packages such as SAS, SPSS, Stata, and so on. But they will be familiar with Excel, and our pedagogical approach is to demonstrate the use of the statistical tools that come either built into Excel or as add-ins that are freely or inexpensively available.

The level of knowledge that is required to get the maximum benefit from this book does not exceed that needed for an introductory statistics course. Therefore, this book is not designed for trained statisticians or PhDs in economics or finance whose education, knowledge, and training far exceed the fundamentals expounded herein.

Is This a Course in Statistics?

The simple answer is no! This book is intended to be an introduction and a “how-to” of some basic statistical techniques that can be useful in a lost profits analysis.  It is not, however, meant to replace a statistical text or give the reader an in-depth understanding of statistics. 

We have provided a glossary of terms as they are defined by standard statistical textbooks, and a bibliography that provides the reader with sources to study for a more in-depth analysis of the concepts introduced in this book. 

While the book focuses on the basic statistical applications as found in Excel or its add-ins, readers are encouraged to undertake a more thorough understanding of the conceptual underpinnings of the techniques by referring to the textbooks recommended in the bibliography.

At a minimum, we suggest the following three Excel add-ins. First, there is the StatPlus add-in that comes with Berk and Carey's book, Data Analysis with Microsoft Excel. Second, there is the popular free downloadable add-in, Essential Regression. And last, if you can find it on the Internet, Gerry LaBute's downloadable add-in, Gerry's Stats Tools. The latter two add-ins come with handbooks that not only serve as instruction manuals for the software, but are primers for regression and statistics in general, respectively.

How This Book Is Set Up

The organizing principle that motivates this book is the attempt to match up Excel's and its add-ins' statistical tools with common, quotidian problems and issues that damages analysts face in their day-to-day practices. We approached the subject matter from both sides of the matchup.

First, we examined the statistical tools available in Excel's Analysis ToolPak, its statistical formulas, and the specialized tools available in the add-ins and asked ourselves: In what ways can we apply any of these tools to commercial damages cases? Second, we reviewed the literature looking for typical commercial damages cases and asked: Is there a statistical solution to this problem? The results of our back and forth approach are the 16 case studies in this book, with each (as the Contents listing shows at the front of this book) presented as its own chapter.

Case Study 1 demonstrates how to use the standard deviation to determine if some number, say, a period's gross margin or a month's sales, falls within an expected range based on past performance.
Case Study 2 concerns itself with testing the sales history of the XYZ Motel to determine if there is an upward trend in the data as asserted by the claimant.
Case Study 3 is an introduction to regression analysis in the context of measuring damages for lost profits as the value of a business destroyed by the actions of the defendant.
Case Study 4 returns to the XYZ Motel and the forecasting of expected sales during the period of restoration using an econometric regression model.
Case Study 5 uses the XYZ Motel data once again to forecast expected sales during the period of restoration using a time series regression model.
Case Study 6 demonstrates the forecasting of sales using an econometric regression model, the determination of saved expenses using a simple linear regression model, and introduces the idea of interrupted time series analysis.
Case Study 7 involves the comparison of pre- and postincident sales and demonstrates techniques to answer the question: Did sales really fall off after the incident?
Case Study 8 demonstrates the forecasting of sales using a time series regression model and tests the significance of an intervening event with the use of interrupted time series analysis.
Case Study 9 involves the issue of cost behavior and estimation.
Case Study 10 presents a problem concerning the determination of saved expenses and introduces the issue of statistical significance vs. practical significance.
Case Study 11 presents the plaintiff's and the defendant's expert's reports in a breach of contract action, points out the flaws in each, and offers a reconciling resolution to their differences.
Case Study 12 is about the application of forensic accounting principles to a lost profits case.
Case Study 13 shows how to set up and use a nonstatistical method for accounting for trend and seasonality when forecasting expected sales.
Case Study 14 involves techniques used to analyze historical sales data searching for trend and seasonality.
Case Study 15 displays nonregression techniques for forecasting sales when the historical sales data is stationary.
Case Study 16 displays nonregression techniques for forecasting sales when the historical sales data is nonstationary.

The Job of the Testifying Expert

According to Federal Rule of Evidence 702, an expert will be allowed to testify in the form of an opinion if,

1. The testimony is based upon sufficient facts or data.
2. The testimony is the product of reliable principles and methods.
3. The witness has applied the principles and methods reliably to the facts of the case.

In addition, the opinion given must be “within a high degree of (economic or financial) certainty.” In other words, a trier of fact, either a judge or jury, is looking for an opinion that will help them to “understand the evidence or to determine a fact in issue.” An academic treatise that increases the storehouse of knowledge might meet that requirement, but given the amount, accuracy, and verifiability of the facts and data available to the expert in a litigation matter, will generally not be forthcoming. Therefore, given the different purposes of the researcher and the testifying expert, different methods of analysis and different uses of the traditional research tools is to be expected.

In the course of this book we will be demonstrating selected statistical techniques to be applied in lost profits cases, where the end result is to form an opinion as to the amount of economic damages, even if there are limits to the facts and data and all the supporting documentation you want is not available. The testifying expert, while using research tools familiar to academics, is attempting to assist the trier of fact, and therefore is not engaged in an “exhaustive search for cosmic understanding but for the particularized resolution of legal disputes.”

About the Companion Web Site—Spreadsheet Availability

There is a companion web site to this book—found at —that contains all the spreadsheets for the case studies in this book. So, you have a choice—you can create the spreadsheets from scratch, following the instructions contained in each chapter, or you can simply download them from the web site and start your analysis immediately. For pedagogical purposes, we recommend that you create your own spreadsheets—there's something about putting them together yourself that leads to a quicker understanding of their purpose.

Adapted from the paper “To Infinity and Beyond: Statistical Techniques Appraising the Closely Held Business,” presented by Drs. Tom Stanton and Joe Vinso at the 20th Annual IBA Conference, San Antonio, TX, January 1998.

Acknowledgments

The authors wish to express their gratitude and appreciation to the following individuals who served as readers and reviewers of this book.

David H. Goodman, MBA, CPA/ABV, CVA
J. Richard Claywell, CPA/ABV, ASA, CBA, CVA, CM&AA, CFFA, CFD, ABAR
John E. Barrett, Jr., CPA/ABV, CBA, CVA
James F. McNulty, CPA

We would also like to thank Nancy J. Fannon, CPA/ABV, ASA, MCBA, for first suggesting the idea of this book and for initially reviewing the introduction and the first six chapters.

INTRODUCTION

The Application of Statistics to the Measurement of Damages for Lost Profits

To get the most out of the case studies in this book, the reader needs to attain a minimum amount of statistical knowledge.

The Three Big Statistical Ideas

There are Three Big Statistical Ideas: variation, correlation, and rejection region (or area). If we can build sufficient intuition about these interrelated concepts, then we can construct a raft for ourselves upon which we can explore the bayou of statistical analysis for lost profits. Therefore, what follows is a very broad introduction to statistics, which does not allow us to explain or define every technical term that appears. To assist you, we have included all those technical terms in a Glossary at the end of the book where they are defined or explained.

Variation

The first Big Idea is that of variation, which means to vary about the average or mean. It deals with the degree of deviation or dispersion of a group of numbers in relation to the average of that group of numbers. For example, the average of 52 and 48 is 50; but so is the average of 60 and 40, 75 and 25, and 90 and 10. While each of the sample data sets has the same average, they all have different degrees of dispersion or variances. Which average of 50 would you have more confidence in—that of 52 and 48, or that of 90 and 10—to predict the population mean?

Variance can also be depicted visually by imagining two archery targets, with one target having a set of five arrows tightly grouped around the bull's-eye and the other target with the five arrows widely dispersed about the target. Not only will the average score of each target be different, but also so will their variances. One could then conclude that based on the widely diverging variances, two different archers were involved.

For statistical purposes, variances are calculated in a specific way. Since some of the numbers in a data set will be less than the average, and hence will have a negative deviation from the mean, we need to transform or convert these negative numbers in some way so that we can compute an average deviation. This transformation consists of squaring each deviation. For example, if the mean is 10 and the particular number is 8, then the deviation is –2. Squaring the deviation gives us 4. Summing the squared deviations of all the numbers in the data set gives us something called, surprise, sum of squared deviations. Dividing this result by the number of observations in the data set minus 1 (n – 1) gives us the sample variance.

Taking the square root of the sample variance produces the sample standard deviation, or the average amount by which the observations are dispersed about the mean.

expresses the relationship among the sum of deviations squared (DEVSQ), variance (VAR), and standard deviation (STDEV).

As we shall see in the case studies, how far a particular number is from 0, a mean, or some other number, measured by the number of standard deviations, comes into play in every parametric statistical procedure we will perform. Nonparametric tests rely on the median, not the average, and therefore have no need of a standard deviation.

Correlation

The second Big Idea is correlation, and to survey that concept we need to go back to the notion of variance and express it in a common-size or dimensionless manner by standardizing the deviations about the mean. For example, in a preceding paragraph we mentioned a mean of 10, a number of 8, and a deviation of –2. If the standard deviation of the data set is 1.5, then by dividing –2 by 1.5 we have standardized, or common-sized, the –2 deviation to be –1.33 standard deviations from the mean. This process would be repeated for each observation in the data set.

For example, assume you have information from 11 purchase and sale transactions that provides you with the selling price of each company's fixed and intangible assets as well as the seller's discretionary earnings (SDE) of each company. To determine the degree of correlation between selling price and SDE, we would first standardize each number in both sets of variables using Excel's STANDARDIZE function. We would common-size the 11 selling prices based on the mean and standard deviation of selling prices. Then we would repeat the process for the 11 SDE values, but using the mean and standard deviation of SDE.

By matching up the corresponding standardized selling price and SDE, we can see how closely they tally with each other. The tighter the match between standardized values, the higher the degree of co-variance, or co-relatedness. To develop a metric that measures the strength of the linear relationship between selling price and SDE, we multiply each set of corresponding standardized values for price and SDE in the Product column, sum the Product column, and then divide by n – 1. The result is known as the coefficient of correlation, symbolized as either r or R. An example is provided in .

Deviations Squared, Variance, and Standard Deviation

Table 0-1

Coefficient of Correlation

Table 0-2

From we can see that the closer the matchup between the variances of the two variables, expressed as the common-sized deviations from their respective means, the higher the correlation coefficient and the stronger the linear relationship between the two variables. This is what is meant when R2 is defined as the metric that measures how much of the variation in the dependent variable is “explained,” or accounted for, or matched up with the variation in the independent variable.

From the above, we can conclude that correlation summarizes the linear relationship between two variables. Specifically, it summarizes the type of behavior often observed in a scatterplot. It measures the strength (and direction) of a linear relationship between two numerical variables and takes on a value between –1 and +1. It is important in lost profits calculations as a summary measure of the strength of various sales and cost drivers.

THE CONCEPT OF THE NULL HYPOTHESIS

An outcome that is very unlikely if a claim is true is good evidence that the claim is not true. That is, if something is true, we wouldn't expect to see this outcome. Therefore, the statement can't be true. For example, a picture of the earth taken from space showing that it is round is not likely to happen if the world is flat. Therefore, that picture is very good evidence that the claim of flatness is not true. Or if the defendant in a criminal trial is not guilty; that is, is no different from or is just like everyone else, then finding gunpowder residue on his hands and the victim's blood on his shoes is strong evidence that he is not innocent, but is in fact guilty of the crime.

These claims, that the earth is flat or not round and that the defendant is not guilty, are called, in statistical terms, null hypotheses. In order to reject these null hypotheses strong evidence has to be produced by the scientist and the prosecutor to convince others to accept the alternative hypotheses, that is, the world is round and the defendant is guilty. In lost profits cases, we too have to deal with claims of null hypotheses—that there is no difference between pre- and postincident average sales; that there is no difference between sales during the period of interruption and the preceding and succeeding periods; or that the slope of our sales forecasting regression line is no different from zero. Rejecting these null hypotheses or accepting their alternatives leads us to the next Big Idea of statistics.

Rejection Region or Area

The final Big Idea is the rejection region or area. In statistics, when we say that we are 95 percent confident of something, we are implying that we are willing to be wrong about our assertion 5 percent of the time. So, how do we set forth the boundaries that measure this 5 percent? The answer to this question flows from our understanding of standardized data. From , we can see that a distribution or list of standardized data has a mean of 0 and a standard deviation of 1, calculated in the same manner as previously described. Assuming that the distribution is also near–bell shaped, or normally distributed, then the empirical rule would come into play, and we would find that about 68.1 percent of the values lie within ±1 standard deviation from the mean and that about 95.3 percent of the values lie within ±2 standard deviations from the mean. This is demonstrated on the bell-curve shown in .

ch03fig001.eps

Therefore, when we choose a confidence level of 95 percent, we are saying that if our test statistic falls within approximately ±2 standard deviations from the mean, or within the 95 percent acceptance area, we cannot reject the null hypothesis, for example, that there is no difference between our number and the sample average, and we must accept the status quo. But if our test statistic falls outside approximately ±2 standard deviations from the mean, or inside the 5 percent rejection area, then we must reject the null hypothesis and accept the alternative hypothesis that things are different because it is very unlikely to find a test statistic and p-value this extreme if there is no difference between our number and the sample average. Therefore, the claim of no difference must not be true and can be rejected.

The rejection region on the right-hand side of the distribution is called the upper tail, and the rejection region on the left-hand side is called the lower tail. If we are asking, for example, if our calculated number is greater than, say, 10, then we would look for our answer in the upper tail alone, prompting the name “one-tailed” test. The same is true if our question about the calculated number concerned it being less than 10. The answer would be found in the lower tail alone, also indicating a “one-tailed” test. However, when we are interested in detecting whether our calculated number is just different from 10; that is, either larger or smaller, we can locate the rejection region in both tails of the distribution. Hence the name “two-tailed” test.

In order to test a null hypothesis we need to create a statistical test that has four elements:

1. A null hypothesis about a population parameter, often designated by the symbol H0. For example, “There is no difference between A and B.” Or, “The difference between A and B is zero, or null.”
2. An alternative hypothesis, which we will accept if the null hypothesis is rejected; often designated by the symbol Ha.
3. A test statistic, which is a quantity computed from the data.
4. A rejection region, which is a set of values for the test statistic that are contradictory to the null hypothesis and imply its rejection.

Two examples of a statistical test are demonstrated in . In each case, the null hypothesis is that the difference between the mean and X is zero, or null; the alternative hypothesis is that there is a difference between the mean and X; the test statistic is (X – mean)/standard deviation, which measures how far X is from the mean as measured in standard deviations; and the rejection region lies beyond 1.96 standard deviations (1.96 is the actual value of the ±2 standard deviations referred to above).

Example 1 Example 2
Mean 100 100
Standard Deviation 25 25
X 135 175
Test Statistic 1.4 3.0
Critical Value 1.96 1.96
Is t-stat > critical value No Yes
Reject null hypothesis? No Yes

The idea of the rejection area will come into play in all of this book's case studies, as we use it to determine whether or not the conclusions of our tests are statistically significant; that is, whether the results happened because of mere chance, or because something else is afoot.

Introduction to the Idea of Lost Profits

Recovery of damages for lost profits can take place in either a litigation setting if the cause of action is a tort or a breach of contract, or under an insurance policy following physical damage to commercial property. In both situations there has been an interruption of the business's revenue stream, causing it to lose sales and eventually to suffer a diminution of its profits. In a tort, lost profits are generally defined as the revenues or sales not earned, less the avoided, saved, or noncontinuing expenses that are associated with the lost sales. For business interruption claims the policy wording is “net income plus continuing expenses.” This is a bottom-up calculation that ought to deliver an equal amount of damages as the top-down calculations used for torts if the fact patterns are the same.

In the top-down approach, the costs of producing the lost sales that do not continue or are avoided or saved might include sales commissions, cost of materials sold, direct labor, distribution costs and the variable component of overhead, or general and administrative expenses. To a damages analyst using this approach, the computation of damages is typically concerned only with incremental revenue and costs, that is, only that revenue that was diminished by the interruption and only those costs and expenses that vary directly with that revenue. The idea of lost profits in a tort or breach of contract situation can be presented schematically, as follows:

Unnumbered Display Equation

Fixed costs are usually ignored as the injured party would generally have to incur those costs regardless of the business interruption.

The schematic for the bottom-up approach, typically used for business interruption insurance claims, is as follows:

Unnumbered Display Equation

Saved, avoided, or noncontinuing expenses are usually ignored as the purpose is to recompense the injured party for the net income they would have earned plus reimbursement for those expenses that continued during the period of interruption, including those such as leases, which might continue due to contractual obligations.

An implication of both these approaches is that the measurement of damages is a multistage process that begins with forecasting sales and then proceeds to indirectly compute lost profits, rather than forecasting lost profits directly. This is so because, as we have explained already, the idea of lost profits is more than the idea of net income—it also includes a component of expense, whichever approach we use. As such, there is no line item on the income statement that is an exact representation of our concept of “lost profits.” Therefore, a lost profits calculation needs to begin with a forecast of expected revenue and then proceed to the examination and classification of expenses into continuing and noncontinuing categories, and then on to those necessary additional steps depending on the chosen approach, before finally arriving at an amount of lost profits.

This section of the introduction will present an overview of the damages measurement process. The second section will introduce various sales forecasting methodologies and will describe the situations that are appropriate for their use.

Stage 1. Calculating the Difference Between Those Revenues That Should Have Been Earned and What Was Actually Earned During the Period of Interruption

Determining what sales would have been during the period of interruption “but for” the actions of the defendant or casualty is the first stage of computing lost profits. For both tort cases and business interruption claims, the damages analyst must rely on a wide range of data and facts to project the expected level of sales. Since the best estimate of the interruption period revenue is related to a variety of factors concerning the capacity to produce and the capability of the market to buy a service or product, the damages analyst needs financial and statistical tools that are capable of incorporating all those factors into a sales forecast. A starting point for measuring the degrees of capacity and capability is to examine what has actually transpired before and after the period of interruption. The business's performance on both sides of the interruption period ought to help identify what the business could have done but for the tort or covered peril, absent other intervening causes. Subtracting actual sales earned from expected sales will produce lost sales or incremental revenues for the period of interruption.

Stage 2. Analyzing Costs and Expenses to Separate Continuing from Noncontinuing

Those costs that vary directly with sales during the pre- or postloss period are good evidence of the saved costs to the firm of not obtaining the “lost” revenues claimed in stage 1. Statistical models can also be useful here to help determine how certain types of costs vary with different levels of service or production. If using a top-down approach, variable costs and expenses should be included in the lost profits calculus, while those that do not vary with sales or production (i.e., fixed or continuing costs) should be excluded from the computation. An example of a variable or saved cost is the income statement line item called “cost of goods sold.” Most or the entire amount of selling expense ought to be variable, and therefore saved, as well. Because financial statement categorization does not necessarily distinguish which costs are variable and which are not, the damages analyst must often use professional judgment and statistical tools to separate continuing from noncontinuing expenses. Regressing costs and expenses on sales can be very effective in this situation if certain requirements are met, as we shall see in future chapters.

In a bottom-up approach, the steps involved are slightly more numerous and include preparing an income statement for the period of interruption that includes all expenses, both fixed and variable. Regression is typically not used in creating the income statement—rather, the analyst's judgment coupled with trends and percentages of sales derived from historical financial statements are used to forecast expected costs and expenses that would have been incurred against expected sales. The next step is to determine what expenses would have continued, based on how costs have behaved in prior periods or actually did continue during the period of interruption.

Stage 3. Examining Continuing Expenses Patterns for Extra Expense

Often certain expenses may increase and new costs may be incurred during or after the business interruption period as a result of the tort or casualty. Management may indicate, for example, that the company had to incur overtime expense to make up lost production, or that temporary office or production space had to be leased. Inquiries of management and the examination of postloss month-to-month changes in wages, overtime, and overhead accounts can identify these costs.

Stage 4. Computing the Actual Loss Sustained or Lost Profits

In the top-down approach, lost profits are the incremental revenues the plaintiff would have earned “but for” the actions of the defendant, less those expenses related to the lost revenues that are saved or avoided. In the bottom-up approach, adding those continuing expenses computed in stage 2 to the expected net income before taxes during the period of interruption and then subtracting any gross profit realized during the same period will give us the business's actual loss sustained. Extra expense incurred by the damaged party resulting from the interruption should be added to the damages. The calculation of each of these elements can be aided with statistical methods.

Choosing a Forecasting Model

The damages measurement scheme not only begins with a sales forecast, but is also, we believe, the critical step in the whole process. Since all the cost and expense considerations that affect lost profits are ultimately dependent on the level of forecasted sales, if we get the sales forecast wrong, even if our expense allocation procedures are correct, the final damages conclusion will still be incorrect. Therefore, we will spend considerable time and effort explaining and in later chapters demonstrating how to get that sales forecast right. To begin, as there are many forecasting models available, how do we choose the most appropriate one for our situation? The one we pick will depend upon five characteristics of the damages measurement question:

1. Type of interruption.
2. Length of period of interruption.
3. Availability of historical data.
4. Regularity of sales trends and patterns.
5. Ease of explanation.

Type of Interruption

Business interruptions can be characterized as “closed,” “open,” or “infinite.”

Unnumbered Display Equation

With a closed interruption, the period of interruption has ended before the damages analyst gets involved. The damages analyst has actual sales data from both before and after the loss period to use in forecasting expected sales.

Unnumbered Display Equation

With an open interruption, the company is still in business, but sales have not yet returned to normal by the time the damages calculations are made. The damages analyst has sales data only from before the loss period to work with and, in addition, will have to determine when the loss period will end, as well as the amount of damages. The question of when to end the period of interruption is as much a legal as a financial issue in a tort, while the typical business interruption policy caps the loss period to the estimated time necessary to rebuild, repair, or restore the damaged property.

Unnumbered Display Equation

An infinite interruption is one where the business suffers through a period of operating losses, then declares bankruptcy or is sold for less than its value at the date of loss. There are only preloss sales data available, and the sales forecast can be used both to compute losses up to the date of sale or bankruptcy and to value the company at the time either of those events takes place. This would be a situation where total losses would entail both a lost profits element and a valuation element. As there would no longer be any cash flows from a business that had ceased operations due to the actions of the tortfeasor, the measure of damages becomes the present value of those lost future cash flows plus the lost profits suffered up to the point of sale or bankruptcy. For a business interruption claim, the same rule applies to an infinite loss as an open loss with the additional limit on the period of interruption to typically not exceed one year.

Length of Period of Interruption

Sales forecasting techniques that are readily applicable to longer-term interruptions, such as multiple months, quarters, or years, are too cumbersome and complicated for short-term losses measured in days or weeks. In those cases, comparison with the same number of days or weeks just prior to the interruption and/or the same time period one year before may be sufficient to determine lost sales as long as there is either no or a minimal upward or downward trend in sales.

Availability of Historical Data

The amount and type of sales data available may force the choice of a forecasting method. If only two or three annual sales figures are obtainable, the options are much narrower than if you have 36, 48, or 60 months' worth of daily, weekly, monthly, or quarterly data. Another consideration is the duration of the loss period—whether it is measured in days, weeks, or months will decide what type of sales data will be needed. A special problem is job shops and construction contractors who record their sales on an irregular basis.

Regularity of Sales Trends and Patterns

Almost all quantitative forecasting methods begin by looking for and discerning patterns in historical sales data, then projecting those patterns into the future as a forecast. The two most important factors of a pattern are trend (upwards or downwards, and straight or curved), and seasonality (e.g., motel sales in the Northeast are high in July and August but lower in January). New products and additional locations can affect sales patterns, and the damages analyst needs to be on the lookout for their appearance in historical sales data, as well as random outliers (abnormally large or small nonrepeating sales figures). A significant amount of noise, or random error, in the sales data will also bear on the choice of forecasting method.

Ease of Explanation

Because the forecasting method and its results might have to be described to someone not familiar with statistical forecasting procedures, such as a judge, jury, or claims manager, the ease with which the method can be explained is important. If you can derive the same sales forecast using multiple techniques, choose the one that is the easiest to explain and understand.

Whichever forecasting model is ultimately chosen, the accuracy of the resulting forecast should be a concern of the damages analyst. In the business world a forecaster can make a projection, wait to see how accurate the forecast turns out to be, and then modify the forecasting model if accuracy is below some acceptable level. Such is not the case in forecasting sales that might have been earned during the period of interruption—those sales will never be observed. However, by performing a “hold back” forecast, which we will demonstrate in one of the case studies, the damages analyst can achieve credibility and demonstrate goodness of fit for the ultimate forecast. In addition, the forecast model should be one that is widely known and used in the litigation support arena.

Conventional Forecasting Models

Keeping the previous cautions and provisos in mind, we can ask which sales forecasting models are likely to work well in lost profits cases. While we have our favorite models, each of the following models has at least one particular situation in which it has proven useful. For now we simply introduce and describe them, while, later in the book, we will present their usage in various case studies.

Simple Arithmetic Models

There are a number of forecasting computations that can be made that involve no more than the four arithmetic functions; for example, computing a simple daily average of the days' sales in any month prior to the injury, and multiplying that daily average by the number of days of expected interruption to arrive at estimated lost sales for the period. Another type of simple arithmetic model is to take the average of the prior and succeeding four weeks' daily sales for those days in question, for example Wednesday to Friday, and use that three-day average as the estimate for lost sales for the missing three days.

Simple arithmetic models are appropriate to use when the period of interruption can be measured in days or weeks and when there is neither a trend nor seasonality present in the historical sales data.

More Complex Arithmetic Models

ex anteex post

The following chapters will demonstrate, through the use of case studies, the application of the various statistical forecasting and analytical models described in this chapter.

Notes

Why divide the sum of squared deviations by n – 1 rather than n? In the first step, we compute the difference between each value and the mean of those values and then square that difference. We don't know the true mean of the population; all we know is the mean of our sample of 11 transactions. Except for the rare cases where the sample mean happens to equal the population mean, the data will be closer to the sample mean than it will be to the true population mean. So the value we compute by summing the squared deviations will probably be a bit smaller (and can't be larger) than what it would be if we used the true population mean in step 1. To make up for this, we divide by n – 1 rather than n. This makes the sample variance a better, unbiased estimator of the true population variance.
But why n – 1? If you knew the sample mean and all but one of the values, you could calculate what that last value must be. Statisticians say there are n – 1 degrees of freedom as we use one degree of freedom to calculate that last value.

For example, to standardize SDE for transaction #1, the formula is ((13,457 – 4,093)/4,005) = 2.34; and to standardize price the formula is ((94,769 – 31,479)/28,524) = 2.22. Their product is 5.19 (2.34 × 2.22).

3. A “hold back,” or ex post forecast is one in which all values of the dependent and independent variables are known and therefore can be used as a means to evaluate a forecasting model. For example, if we construct a forecasting model based on 36 months of historical sales data, we can test the accuracy of that model by “holding back,” say, the last five months and input 31 months of those sales back into the model to forecast the “held back” five months. We would then compare the fit of the actual sales to the predicted sales for the five months in order to assess the forecast's accuracy.