< PreviousA PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES 10 SEPTEMBER | OCTOBER 2020 t h e v a l u e e x a m i n e r •Markets, market share, and key competitors and suppliers • Business risks and government regulation •Strategy and future plans From the company’s website, interviews with management, and observations at the site visit, I fill in what I learned and then ask the client to review my understanding and add any missing information. I advise them that this is their story and the more details they can provide, the better understanding the trustees and advisors will have of the business. That will aid them in determining the fairness of the value. Economic and Industry Conditions At the end of this section, I summarize key points under the header “How Economic and Industry Conditions Affect the Value of the Company” to help the trustee and other users. This narrative includes factors that would impact the growth rate used in the valuation and other industry conditions that affect risk. Examples include a lobster wholesale distributor whose export channel to China is halted due to tariffs, a home builder affected by mortgage interest rates, or a construction company whose ability to hire workers is affected by low unemployment rates. Historical Financial Statements I ask management to provide a list of expenses that will not be incurred once the company is an ESOP company. These expenses might include unnecessary travel and entertainment, compensation paid to family members who will not continue to work in the business, or professional fees incurred in connection with the ESOP transaction. The financial statements are normalized with these adjustments. Future management salaries may be limited as part of the transaction. If so, management compensation should be restated. Nonoperating assets and liabilities should be identified, and the analyst needs to know whether these items will continue to be held by the company after the ESOP transaction. Examples include land held for investment, an investment in a joint venture, or shareholder loans. Analysis of Historical Financial Statements At the end of this section, I include a header, “How Industry Comparison Affects the Valuation.” In addition to working capital adequacy, this summary discusses profitability, leverage, accounts receivable, and inventory turns and current and quick ratios as compared to industry, and includes a conclusion as to whether the company appears weaker or stronger than companies in the industry. Valuation Conclusion The report should be clearly written so that the trustee can understand the assumptions and conclusions. The analyst should bear in mind that the report is subject to review by the DOL and that such review, if it happens, likely will take place several years after the transaction date. There have been many The analyst should bear in mind that the report is subject to review by the DOL and that such review, if it happens, likely will take place several years after the transaction date.A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES t h e v a l u e e x a m i n e r SEPTEMBER | OCTOBER 2020 11 lawsuits and investigations of ESOP transactions. The NCEO has published a guide7 and produced webinars on the issues involved. The values produced by the various approaches should be presented and the rationale for the conclusion of value should be discussed in detail. To assist the trustee and the trustee’s advisors in assessing the fairness of the value obtained, I include a sensitivity analysis after the conclusion of value. Below is an example: Sensitivity Analysis The conclusion of value was $2,230,000 as a result of capitalizing the Company’s EBITDA. The factors that affect this conclusion of value, as explained in the narrative above, are: Cost of debt capital 1.96% Cost of equity capital 17.63% Growth rate 4.0% Blended tax rate 24.0% 2019 EBITDA $360,473 To provide additional information to the trustee and the firm providing the fairness opinion, we computed the value of the Company by changing one of the above factors and leaving the other factors unchanged, as described below. The calculated values range from a low of $2,065,000 to a high of $2,355,000. If the average of 2018 and 2019 EBITDA were used, the value produced by capitalizing EBITDA would be $2,065,000. If a 3 percent growth rate were used, the value produced by capitalizing EBITDA would be $2,109,000. If a 5 percent growth rate were used, the value produced by capitalizing EBITDA would be $2,270,000. If a 19 percent cost of equity capital were used, the value produced by capitalizing EBITDA would be $2,080,000. If a 17 percent cost of equity capital were used, the value produced by capitalizing EBITDA would be $2,355,000. The ESOP Transaction Once the report is finished, I send a draft to the trustee and the trustee’s advisors so that the conclusion of value can be used in determining the purchase price for the stock. Once that is decided, a paragraph is included in the report following the sensitivity analysis that describes the transaction, and the report is issued. The example below comes from a transaction in which the company redeems the shareholder immediately prior to the ESOP stock acquisition and the acquisition debt leads to a substantial decline in equity value. 7 Michael Coffey et al., A Guide to DOL ESOP Investigations (Oakland, CA: NCEO, 2019) (available for purchase To assist the trustee and the trustee’s advisors in assessing the fairness of the value obtained, I include a sensitivity analysis after the conclusion of value.A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES 12 SEPTEMBER | OCTOBER 2020 t h e v a l u e e x a m i n e r The ESOP Transaction The transaction, as proposed, consists of the Company purchasing 100 percent of its currently outstanding shares in a redemption from its current shareholder, XXX. The purchase price for the redemption will total $XXX. The purchase will be financed by a note payable over 15 years with interest at 2 percent. The note will require interest-only payments for 2020 through 2035 and principal payments required in 2035. Payments may be made earlier if the Company chooses to pay or the holder requests payments subject to a working capital cushion. Following the redemption, the Company’s equity value will drop to $XXX, and the Company will immediately issue 100,000 shares of stock to the Company’s ESOP, which will be formed to take effect on X/X/2020. The purchase price will be $XXX. The consideration will be a purchase money note in the amount of $XXX from the ESOP trustee to the Company, with the ESOP shares pledged as collateral to the Company. It will have a term of 40 years and carry an interest rate of 4 percent. No Material Change Letter As mentioned above, the trustee and the trustee’s advisors may request the valuation analyst to prepare a no material change letter covering the period from the valuation date to the closing date. Depending on the valuation date, it may not be possible to issue this letter if the effects of COVID-19 on the company’s operations cannot be determined. The procedures to be performed are at the discretion of the analyst. Typically, these include a discussion with management and obtaining current financial statements. Balance sheets at the valuation date are compared to the balance sheet nearest to the closing date to look for material changes, such as major fixed asset purchases, new loans, and decreases in cash, accounts receivable, and inventory. The interim income statement is reviewed for any unusual items. Any such changes and the reasons for them are discussed in the letter. I request a representation letter from management. A sample representation letter from a mechanical contracting company is presented in the box on this page. In connection with the valuation of XYZ Inc. as of December 31, 2019, to be used for the ESOP transaction, we hereby represent the following as of March 17, 2020 (closing date). 1. Our contracts-in-progress schedule is only updated quarterly. We monitor job costs, and there are no significant downward changes to the estimated gross profit on our jobs in process. 2. Due to the effects of COVID-19, jobs may be slowed or closed for a period by the general contractors. However, as of this date, our employees are working on our jobs in process. We have applied for a Paycheck Protection Program loan. 3. Our service department is operational and doing business as usual. 4. There has been no material change in XYZ Inc.’s financial condition since December 31, 2019 that would cause a decline in the value of the Company. In making this representation, we considered the following: •We have no knowledge of any legal actions filed against the Company. •There were no shareholder distributions since December 31, 2019. • There were no extraordinary expenses other than those for the ESOP transaction. •There were no major capital expenditures since December 31, 2019. •There were no new loans since December 31, 2019. Payments on existing loans were made as required. Wording for the No Material Change Letter The attorney advising the trustee will communicate his or her required wording for the letter. The letter is typically two to three pages and usually includes the following: Depending on the valuation date, it may not be possible to issue this letter if the effects of COVID-19 on the company’s operations cannot be determined. Sample Representation LetterA PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES t h e v a l u e e x a m i n e r SEPTEMBER | OCTOBER 2020 13 •A statement that the trustee requested the analyst to undertake an independent analysis to determine if there has been a change in the valuation of the common stock between the analysis date and date of closing. •A statement that the valuation was conducted according to IRS Revenue Ruling 59-60, the DOL proposed regulations (29 CFR 2510.3-18), and the analyst’s professional standards (AICPA SSVS, NACVA, etc.). •Definition of fair market value. • Discussion of what procedures were performed and what the findings were. Transaction Closing At the transaction closing, the analyst will be asked to present the report and his or her conclusions. The trustee and the trustee’s advisors can ask further questions about assumptions and data. The trustee will document this presentation as part of the minutes of the closing. Conclusion This article has discussed some of the factors an analyst should consider when undertaking a valuation for an ESOP transaction. The client is the ESOP trust to be formed, not management. The report should refer to the (withdrawn) DOL regulations. The analyst should write the national and local economic, industry, and industry comparison sections to allow the trustee to understand how these factors affect the conclusion of value, and should fully explain any normalizing adjustments. The analyst should also describe the factors used to determine the specific company risk factor in the cost of equity capital, and write the conclusion of value in a manner that allows the trustee to follow the analyst’s rationale. Including a sensitivity analysis helps the trustee in his or her decision process. If the analyst is asked to prepare a no material change letter, he or she should carefully analyze the data and obtain a representation letter from management. When concluding as to the value of the interest, I always ask myself, “Would I buy this business for this amount? How will the proposed debt and payout terms affect the company’s ability to continue operations?” These valuations can be high risk, so the analyst should ensure that the report complies in all aspects with professional standards and is written in such a way that the trustee and other users, including the DOL, can understand the assumptions and conclusions. Judith H. O’Dell, CPA, CVA, is president of O’Dell Valuation Consulting LLC CPA. She obtained her CVA designation in 2000 and has performed valuations for ESOP, divorce, estate, gift, and litigation purposes. Ms. O’Dell serves on the editorial board of The Value Examiner. She is the former chair of FASB’s Private Companies Financial Reporting Committee and served on the Audit Advisory Committee of the U.S. Government Accountability Office, as trustee of the Financial Accounting Foundation, and on the AICPA board of directors. Ms. O’Dell holds a BA in economics from Immaculata University and an MFA in visual storytelling from Maine Media College. Email: VE These valuations can be high risk, so the analyst should ensure that the report complies in all aspects with professional standards and is written in such a way that the trustee and other users, including the DOL, can understand the assumptions and conclusions.A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES 14 SEPTEMBER | OCTOBER 2020 t h e v a l u e e x a m i n e r VALUATION /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// The Transaction Databases: Some Statistical Questions and Concerns By Mark G. Filler, CPA, ABV, CVA, AM, CBA In life in general and in the world of valuation in particular, we often take as an article of faith those ideas that should always be testable hypotheses. For example, we always assume that there is a causal, or at least a highly correlative, linear relationship between price and earnings (however defined) or between price and market share or between price and relative profitability. We also assume that the coefficient of variation is a valid selection tool to choose among valuation metrics. And lastly, we assume that when we regress price against earnings, market share, or relative profitability that a high R-square, a low standard error of the estimate, and a p-value of less than 0.05 tell us all we need to know about the validity of the model. I recently explored the use of the Bizcomps, IBA, and DealStats transaction databases for the used car industry (NAICS code 441120 and SIC Code 5521) and found that testing these three assumptions led to some surprising results for all three databases. What follows is an exploration of those results along with some statistical rules of thumb to follow when using the databases. Before a valuation analyst can predict price based on sales, seller’s discretionary earnings (SDE), gross profit, etc., the analyst needs to answer two basic questions: Is there a statistically significant linear relationship between the two variables? And even if there is, does it have any explanatory power? In other words, is the coefficient of determination, r2, greater than 0.50 so that the valuation analyst can opine to a concluded value with a reasonable degree of certainty that the opinion will pass the “more likely than not” test? The Coefficient of Correlation When performing a valuation using the direct market data method, a basic requirement is that there be not only a logical relationship between price and sales (or EBITDA or SDE, etc.) but also that it be, at a minimum, a statistically significant linear correlated relationship, as the same basic assumptions for linear regression also apply to correlation. Of course, linearity is a necessity for a useful linear regression model, but it is also essential for ratio models, no matter the measure of central tendency used—mean, median, or weighted harmonic mean of the chosen ratio. And hand-in-hand with linearity is the idea that price, the dependent variable, ought to be, at a minimum, symmetrical about its mean. Since most of the sample sizes we derive from the transactional databases are small, we cannot assume that the central limit theorem will overcome the infirmity of nonsymmetrical samples as it does when sample size is at least 30, and even better, when it exceeds 50. Therefore, we need to test for symmetry for the dependent variable (the independent variable need not be symmetrical) before doing a correlation test. If the dependent variable, price, is at least symmetrical, if not near-bell-shaped, and at best, normally distributed, we can use Pearson’s product-moment correlation coefficient, denoted as Pearson's r, to test for correlation, and by substitution, also test for linearity. Defaulting to Spearman’s1 rank correlation coefficient, denoted as s, is not a solution, as it does not measure linearity, but association. Assuming the dependent variable passes the symmetry tests, the results of the Pearson correlation test will tell us to what degree the variables are either positively or negatively linearly correlated with each other, using a scale from -1 to +1, where absolute 1 is perfect correlation. If we wish to make inferences about the population or test the hypothesis that there is no linear relationship between the two variables, we can then assess the coefficient of correlation for statistical significance with a t-test. If our null hypothesis is that there is no linear relationship between price and SDE, 1 All statistical calculations were made using either Excel’s built-in functions, or the functions and tools of StatPlus, the Excel add-in that comes with Berk and Carey’s introductory text, Data Analysis with Microsoft Excel, or the functions or tools of Real Statistics, the Excel add-in available for free at A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES t h e v a l u e e x a m i n e r SEPTEMBER | OCTOBER 2020 15 for example, then our rejection of the null hypothesis will depend on the size of the correlation coefficient, the number of observations in the sample, and the percentage of test results we are willing to accept as incorrect, e.g., 10 percent, 5 percent, or 1 percent or less. The resulting p-value of the t-test is just the first step. If the p-value is less than, say 0.05, then we have a statistically significant linear correlation coefficient. We next have to inquire about the explanatory power of the correlation model. If, for example, we have a correlation coefficient of 0.625 with a p-value of 0.021, indicating a linear relationship, we can test explanatory power by calculating an r2, or coefficient of determination, of 0.391. This means that the variation in the independent variable, sales or SDE, explains only 39.1 percent of the variation in the dependent variable, price. This leaves 60.9 percent of the variation unexplained, and puts the valuation analyst, as a testifying expert, in an untenable situation. The Coefficient of Variation To use the unitless measure of dispersion called the coefficient of variation (standard deviation ÷ mean), two criteria must be met. First, the distribution of price-to-sales, etc., must be at least symmetrical, if not near-bell-shaped, and at best, normally distributed. Second, the coefficient of variation must be statistically significantly different from zero, which can be calculated using a t-test. As the test for symmetry is the more important of the two tests (in the valuation context, we are always looking for coefficients of variation that are closer to zero than not), how do we determine that the distribution meets the minimum requirement of symmetry? The following are indicative of such: • A box plot that is relatively symmetrical; i.e., the median is in the center of the box and the whiskers extend equally in each direction •A histogram that looks symmetrical •A mean that is approximately equal to the median, which is approximately equal to the weighted harmonic mean • A coefficient of skewness that is relatively small; i.e., between -1.25 and +1.25, where perfect skewness is zero2 2 For a small sample of n = 15, the standard error of skewness is 0.6325, and the critical value is 1.96 × 0.6325 = 1.24. Hence, if the absolute skewness value is less than 1.25, we can assume symmetry. As the sample size increases—and, per the central limit theorem, the sampling distribution of the mean becomes more normally distributed—the critical value decreases to compensate. If the distribution is classified as normal by the Shapiro- Wilk, D’Agostino-Pearson, and Lilliefors tests, then it is, by definition, symmetrical and no further analysis is required. Even if the distribution of valuation ratios meets these two criteria, a broader question is whether the coefficient of variation conveys any useful information if the numerator and denominator of the valuation ratio are either not significantly correlated or lack any explanatory power. Masking in Linear Regression Frequently in regression analysis applications, the data set contains some cases that are outlying or extreme; that is, the observations for these cases are well separated from the remainder of the data. These outlying cases may involve large residuals and often have dramatic effects on the fitted least squares regression function. It is therefore important to study the outlying cases carefully by means of visual tests, such as scatter plots and residual plots, and computational tests, such as standardized residuals, and decide whether they should be retained or eliminated. I have written previously about removing outliers whose residuals are more than 2.5 standard deviations from the mean as these transactions are, by their nature, dissimilar from and not relevant to the subject company.3 But what about those extreme cases that are not outliers as defined in the previous paragraph, as they lie on the plane of the regression line and therefore have small residuals, but instead are potential influencers and/or leverage points as they are separated from the range of all the other observations? These cases mask, or cover up, the effects of the bulk of observations, such as outliers described above, and hence, have a great deal of influence on the totality of the regression output. In fact, as we shall see, a leverage point can turn a poorly performing regression model into one that appears to have great promise. We can test for influence or leverage with statistical tools or we can simply look at a scatter plot to identify them. Once identified, they can be removed from the model inputs to see to what extent the regression output changes. 3 Mark G. Filler, “Is There a Buy-a-Job Phenomenon in Business Valuations?,” Valuation Strategies 7, no. 6 (July/August 2004): 20-33.A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES 16 SEPTEMBER | OCTOBER 2020 t h e v a l u e e x a m i n e r Applications and Examples Ratio Models Valuation analysts typically select a particular array of ratios, assume that it is not symmetrical, and select the median as their measure of central tendency. Or, they will remove the outliers from the ratio array, thereby making it symmetrical, and use either the arithmetic mean or the weighted harmonic mean as their measure of central tendency. In either case, the analyst has assumed what he or she needs to prove—that the numerator and denominator of the ratio are linearly correlated and have sufficient explanatory power. In the balance of this article, we will walk through an outline of the exploratory steps necessary to test these two assumptions. Using the aforementioned used car dealer transaction data, we will start with the price- to-sales ratio, with input data and descriptive statistics shown on Exhibit A.4 We began with 53 asset sale transactions from the three databases: 24 from DealStats, 17 from IBA, and 12 from Bizcomps. Of the 53 transactions, there were eight duplicates, or 15.1 percent of the total, and of those eight, three were in each of the databases, while five were in only two. Since we were not combining the databases, this low level of duplication should not distort the results of our inquiry. Then we tested for outliers greater than 2.5 standard deviations from the mean and removed one transaction each from the DealStats and IBA databases. Next, we tested the ratio arrays for symmetry, using the coefficient of skewness and the Shapiro-Wilk, D’Agostino-Pearson, and Lilliefors tests. Only the IBA ratio array passed this test. Table 1 Price to SalesDealStatsBizcompsIBA Ratios SymmetricalNoNoYes Near-Bell-ShapedNoNoYes NormalNoNoNo Price SymmetricalNoYesNo Near-Bell-ShapedNoYesNo NormalNoYesNo Pearson’s r 93.81%41.85%64.91% p-value .0000.1757.0065 r2 .880.175.421 4 Exhibits A through J may be found online A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES t h e v a l u e e x a m i n e r SEPTEMBER | OCTOBER 2020 17 While symmetry,5 at a minimum, is required for correlation testing, and two of the databases fail this basic test, even if we impute symmetry,6 we can see from Table 1 that the Bizcomps data set is not statistically significant, with a p-value greater than 0.05. Nor does it have any explanatory power, even with a normally distributed dependent value, with an r2 of 0.175. The IBA data set is near-bell-shaped and statistically significant, with a p-value less than 0.05, but it too lacks explanatory power with an r2 of 0.421. Again, ignoring lack of symmetry, only the DealStats data set demonstrates both statistical significance and strong explanatory power. These results appear to be too good to be true for nonsymmetrical data, and in fact they are a consequence of masking, which we will explore later in the regression section of this article. But for now, as the DealStats and Bizcomps ratio data sets are not symmetrical, we cannot use average measures of central tendency from either of them. The IBA ratio data set is symmetrical, which would allow us to use the average or the weighted harmonic mean as a valuation metric if the price-to-sales relationship had any explanatory power. If the Bizcomps data set had a low enough p-value and a high enough r2, the median would be available as a measure of central tendency. But it does not. Therefore, we are left, for now, with the DealStats median price-to- sales ratio as our default measure of central tendency, which is shown on Exhibit A in the DealStats ratio column as 0.141. Now, let us turn our attention to the price-to-SDE data sets, with input data and descriptive statistics shown on Exhibit B7 for the Bizcomps and IBA databases (DealStats only had nine SDE transactions, a number deemed insufficient for this exercise). Initially, we tested for outliers greater than 2.5 standard deviations from the mean, and removed one and two transactions, respectively, from the IBA and Bizcomps databases. Next, we tested for symmetry using the coefficient of skewness and the Shapiro-Wilk, D’Agostino-Pearson, and Lilliefors tests. For this model, both the price and ratio arrays are more than just symmetrical, they are normally distributed, allowing us to utilize Pearson’s correlation coefficient, which results are shown in Table 2 along with the r2 statistic. Ignoring lack of symmetry, only the DealStats data set demonstrates both statistical significance and strong explanatory power. 5 As Shapiro-Wilk is the more powerful test for normality, if the data array shows normality under it, the data array will be deemed to be normal. If the coefficient of skewness is between -1.25 and +1.25, and the data array passes none of the normality tests, it will be deemed to be only symmetrical. If the data array is symmetrical and passes one or both of the weaker D’Agostino-Pearson and Lilliefors normality tests, it will be deemed to be near-bell-shaped. 6 Many statistical tests require that the data be normally distributed. However, most of these tests are quite robust to violations of normality, especially when the data is at least symmetrically distributed. Being symmetrical about the mean allows the empirical rule to be invoked, which explains that approximately 68 percent of the data points will lie within one standard deviation of the mean, about 95 percent within two standard deviations of the mean, and about 99.7 percent within three standard deviations of the mean. 7 See A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES 18 SEPTEMBER | OCTOBER 2020 t h e v a l u e e x a m i n e r Table 2 Price to SDEBizcompsIBA Ratios SymmetricalYesYes Near-Bell-ShapedYesYes NormalYesYes Price SymmetricalYesYes Near-Bell-ShapedYesYes NormalYesYes Pearson’s r54.01%78.96% p-value.0863.0066 r2.292.623 Even though the array of price-to-SDE ratios is normal for both databases and the dependent variable (price) is normally distributed, the Bizcomps relationship between price and SDE is neither statistically significant at the 0.05 level, nor does it meet the threshold level of explanatory power, leaving us unable to use any of the price-to-SDE measures of central tendency to formulate a price for a subject company. The IBA price-to-SDE relationship is both statistically significant and above the threshold of explanatory power, as we would expect with the ratio array and the dependent variable both normally distributed. But, as we shall later demonstrate, the IBA r2 of 0.623 and the p-value of 0.0066 for Pearson’s r are artifacts of masking. Finally, let us investigate three valuation metrics exclusive to the DealStats database: price-to-gross profit, price-to-EBIT, and price-to-EBITDA, with input data and descriptive statistics shown on Exhibit C.8 Initially, we tested for negative earnings and outliers greater than 2.5 standard deviations from the mean and removed one EBITDA transaction and two transactions each from the EBIT and gross profit arrays. Next, we tested for symmetry using the coefficient of skewness and the Shapiro-Wilk, D‘Agostino-Pearson, and Lilliefors tests. For these three models, the dependent variable (price) is not even symmetrical, never mind normally distributed, which will have an impact on the regression models that we will inspect shortly. However, all three models have symmetrical ratio arrays, allowing us to utilize Pearson’s correlation coefficient, which results are shown in Table 3 along with the r2 statistic. For these three models, the dependent variable (price) is not even symmetrical, never mind normally distributed, which will have an impact on the regression models. 8 See A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINES t h e v a l u e e x a m i n e r SEPTEMBER | OCTOBER 2020 19 Table 3 DealStatsPrice to Gross ProfitPrice to EBITPrice to EBITDA Ratios SymmetricalYesYesYes Near-Bell-ShapedYesYesYes NormalYesNoYes Price SymmetricalNoNoNo Near-Bell-ShapedNoNoNo NormalNoNoNo Pearson’s r99.91%22.04%-1.46% p-value.0000.3243.9572 r2.998.049.000 Once again, even though the array of price-to-earnings ratios is symmetrical for all three databases, the relationships between price and EBIT and EBITDA are not statistically significant and neither meets the threshold level of explanatory power, leaving us unable to use any of the price-to-EBIT or price-to-EBITDA measures of central tendency to formulate a price for a subject company. Price-to-gross profit appears to tell a different story: the array of ratios is normally distributed but the dependent variable price is not even symmetrical. However, the dependent and independent variables are correlated with a high degree of statistical significance, and with an r2 of 0.998 the model has much in the way of explanatory power. But as we shall see, this too is an artifact of masking. Regression Models We will now turn our attention to the masking issue that is presented in matched pairs of scatterplots: Exhibits D and E, F and G, and H and I.9 Each of the masked charts—D, F, and H—show an observation that, while lying on the plane of the trend line, is at a great distance from the central group of observations and, therefore, is a leverage point. As the trendline lies close to the leverage point, it has a small residual and therefore does not show up as an outlier. The question is: Is the leverage observation news or noise? Does it tell us anything about a pricing multiple that we should consider for our subject company? Before we answer that question, let’s examine the unmasked charts—Exhibits E, G, and I—to see if the regression outputs have changed significantly after removing the leverage point from the data set . To that end, let us compare regression output metrics for the paired data sets. For Exhibits D and E DealStats—Price to Salesr2t statisticCoVMedian Sales Value Masked .88016.4134.7%($155,272) Unmasked .09700.074.2%$193,000 9 See Next >