A Quantitative Approach to Commercial Damages: Applying Statistics to the Measurement of Lost Profits

By Mark G. Filler and James A. DiGabriele

How-to guidance for measuring lost profits due to business interruption damages

A Quantitative Approach to Commercial Damages explains the complicated process of measuring business interruption damages, whether they are losses are from natural or man-made disasters, or whether the performance of one company adversely affects the performance of another. Using a methodology built around case studies integrated with solution tools, this book is presented step by step from the analysis damages perspective to aid in preparing a damage claim. Over 250 screen shots are included and key cell formulas that show how to construct a formula and lay it out on the spreadsheet.

• Includes Excel spreadsheet applications and key cell formulas for those who wish to construct their own spreadsheets
• Offers a step-by-step approach to computing damages using case studies and over 250 screen shots

Often in the course of business, a firm will be damaged by the actions of another individual or company, such as a fire that shuts down a restaurant for two months. Often, this results in the filing of a business interruption claim. Discover how to measure business losses with the proven guidance found in A Quantitative Approach to Commercial Damages.

• Language: English
• Publisher: Wiley
• Release date: Apr 9, 2012
• ISBN: 9781118236376

Book preview

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.¹

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.

Table I.1 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 Table I.2.

TABLE I.1 Deviations Squared, Variance, and Standard Deviation

Table 0-1

TABLE I.2 Coefficient of Correlation

Table 0-2

From Table I.2 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 Table I.2, 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 Figure I.1.

Figure I.1

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 Table I.3. 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).

TABLE I.3

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.,