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2 measured by earnings before interest, taxes, depreciation and amortization, or EBITDA ). The appraiser must decide what measure(s) (the independent variables) such as revenues, EBITDA, profit margin, etc. best predict the value (the dependent variable, y) of the private company. Therefore, using Excel the analyst performs several linear regressions to test the relationship between the prices paid (y) for the beer distributors that were sold and each of the independent variables (the annual revenues, EBITDA and profit margin of each beer distributor) to discern relationships the price paid. Shown below is a summary of the data for each acquired company, including the price paid, the price expressed as a multiple of annual EBITDA and revenues, and the annual EBITDA and annual revenue figures of each beer distributor: Prices Paid to Acquire Beer Distributors (\$ in 000s) Price Paid as Multiple of: Annual Price Paid EBITDA Revenues EBITDA Revenues \$94, \$13,457.0 \$61,283.0 \$52, \$7,448.0 \$62,444.0 \$48, \$5,884.2 \$61,508.2 \$47, \$5,845.6 \$60,119.3 \$33, \$2,888.1 \$42,705.0 \$33, \$4,600.0 \$32,000.0 \$21, \$2,854.8 \$26,674.2 \$7, \$1,087.9 \$8,704.1 \$3, \$132.6 \$5,901.2 \$2, \$661.0 \$3,144.0 \$1, \$165.7 \$3,138.7 Average Median tionships using simple linear regression, first regressing the price paid (the y, or dependent variable) against EBITDA (the x, or independent variable), then do the same thing with price compared to revenues. Therefore, our formulas are as follows, first based on earnings and then revenues: Formula of Relationship Between Price Paid and a Selling Company s Earnings (EBITDA) in Transactions where y = Company value (in \$000s)- this is the price we are trying to predict x = annual EBITDA (in thousands)- This is the earnings of the company, i.e., its earnings before interest, taxes, depreciation and amortization expense a= slope intercept Formula of Relationship Between Price Paid and a Selling Company s Annual Revenues in Transactions where y = Company value (in \$000s)- this is the price we are trying to predict x = annual revenues (in thousands) a= slope intercept Regression analysis works by plotting the independent and dependent variable data from the transactions on a graph and then mathematically draws a line between through the very center of the data, minimizing the squared deviations of the data from the average. The formula then describes this relationship so that it might later be applied to the private company s specific results to estimate its value. Shown below is the data on the price paid for the acquired beer distributors versus their annual earnings (EBITDA) (results in \$000s): The valuator could just simply take the easy Price Paid (y) EBITDA (x) way out and take the median multiple as a measure of \$94,769.0 \$13,457.0 the central tendency in the price paid, and in many \$52,000.0 \$7,448.0 instances that might be the right decision. However, \$48,400.0 \$5,884.2 \$47,000.0 \$5,845.6 the data above shows a wide range of multiples, so \$33,740.0 \$2,888.1 the appraiser wants to be sure he or she is making the \$33,715.0 \$4,600.0 best informed decision. Second, it would be very helpful \$21,100.0 \$2,854.8 to know which multiple is a better predictor of \$7,500.0 \$1,087.9 what buyers of beer distributorships pay in acquisitions, \$3,700.0 \$132.6 one based on earnings (EBITDA) or revenues. \$2,550.0 \$661.0 \$1,800.0 \$165.7 Insight into this issue will assist in deciding which multiple(s) to use or weight the most in the valuation. Neither of these questions is easily answered from merely eyeballing the data, so let s test the rela- (Continued on Page 3) 2

4 acquired companies. Shown below is the data on the price paid for the acquired beer distributors versus their annual revenues (results in \$000s): Inputting the same data in Excel and running a simple regression provides the following results: Price Paid (y) Revenues (x) \$94,769 \$61,283.0 \$52,000 \$62,444.0 \$48,400 \$61,508.2 \$47,000 \$60,119.3 \$33,740 \$42,705.0 \$33,715 \$32,000.0 \$21,100 \$26,674.2 \$7,500 \$8,704.1 \$3,700 \$5,901.2 \$2,550 \$3,144.0 \$1,800 \$3,138.7 In this instance, we can quickly see two things. First, the R square measure of tells us that 72.6% SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 9 ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Revenues (x) of the variation in the prices paid for the beer distributors sold can be explained by changes in their level of annual revenues. While 72.6% is a relatively strong relationship, it is not nearly as powerful a predictor as with earnings (EBITDA), which explained 97.5%. Nonetheless, since the Significance F figure is less than 0.05, we can conclude that the relationship is not random and is statistically significant. Given these results, the appraiser might then proceed to use the predicted regression formula based on earnings (EBITDA) to value the private company at issue using as follows, inserting the private company s measures to arrive at a value estimate as follows: As shown above, using the regression formula, the estimated value for the company is \$16.7 million. Applying Merger and Acquisition Data Regression Formula to Estimate Company 100% Control Value (\$000s) Multiple applied (see previous section) (b) Times: Company EBITDA \$2,000.0 Equals: \$14,000.0 Plus: Y Intercept (a) 1 \$2,686.3 Equals: Preliminary Total Value of Invested Capital (TIC) \$16,686.3 Less: Interest-Bearing Debt 2 \$0.0 Equals: 100% Control Value of Company Equity \$16, Based on regression obtained from the analysis of prior transaction data. The formula based on EBITDA is as follows: y= a + bx, y = \$2, x. 2 This is a simplistic example, where the price paid from the transaction data was defined as the total price including any interest-bearing debt assumed. Therefore, to arrive at the value of the Company s common equity, we have to subtract interest-bearing debt (it had none). Adjustments, if any, to be made in a real valuation will depend upon the nature of the data used. Caution in Interpreting Results Several major warnings are in order: 1. Regressions do not prove causality- a cause and effect relationship. A relationship could have a high R square and also be shown to be statistically significant. Nonetheless, the result could be utterly meaningless and there may in fact be no casual relationship between the independent variable and dependent variable. This is why commonsense has to also play a (Continued on Page 5)

5 role in assessing the results of statistical analysis. In this instance, the results make sense- companies that have higher earnings sell for more, and this also fits well with the traditional foundation of valuation theory that the value of a company is based on the present value of its earnings or cash flow. 2. Not all relationships are necessarily linear in nature, so in those instances tools other than regression analysis may be indicated. Similarly, regressions are based on an assumption that the sample population data is normally distributed, i.e., like the bell-shaped curve. When the population is not normally distributed regression might or might not be appropriate depending upon the circumstances. 3. The quality, quantity and reliability of the underlying data impacts the degree to which regression analysis can be used and whether or not it gives reliable results. A practical problem in business valuation is that there is often insufficient data to have a large enough sample to be able to effectively utilize regression analysis. 4. While it is appealing to have the simplicity of a math formula to predict value, valuators must not lose sight that other non-quantitative measures and issues may be equally important in impacting value. As a simple example, the valuator finds out that the beer distributorship being valued is about to lose its exclusive right to distribute a national beer brand, the very thing that drove its revenues and earnings in the first place. Active-Passive Appreciation Uses of Regression A business appraiser has been engaged to value a distributor of residential wood flooring (Woodco) in a divorce for the husband who owns and runs it and who had owned the shares of the company prior to getting married, therefore being his separate property. The appraiser has been given the additional task of determining how much of the change in the value of the company during the marriage (from the time of marriage to the date the parties separated) was due to the active efforts of the husband, versus the amount of the change in value due to passive, external forces. Under relevant state law, even though the shares were the husband s separate property and not subject to division on divorce, the portion of the change in the value during the marriage that is a result of his active efforts is considered marital for purposes of equitable 5 distribution. Hence, the need to isolate the impact of passive, external factors on the change in value, versus those that are active. There are many potential factors external to Woodco that might impact its performance and its value at different points in time. The valuator s research and analysis has isolated what he believe are three passive forces affecting Woodco s results and it change value over time: Changes in the level of residential housing starts Changes in investor required rates of return for investing in common stocks Changes in income tax rates, personal and corporate Analysis of each of these passive forces will be considered by the valuator, although, for brevity sake, this article will only examine the effect of housing starts. Since the parties were married on January 1, 1989 and were separated on December 31, 2006, the valuator will examine the relationship between housing starts and Woodco s performance over that time frame. Housing Starts as a Passive Force2. Findings of Relationships- Housing Starts to Revenues Based on the interview of Woodco officials as well as a review of public company filings with the Securities and Exchange specializing in the sale of various building materials, it appears that the level of single family residential housing starts is a significant external factor affecting results in a given year. This is a factor over which neither the husband nor other Woodco management has any control and is therefore a passive force. The overall level of residential new home construction is influenced by changes in interest rates, the overall economy, consumer confidence, population growth, the rate of new household formations, and other factors. Woodco s products are ultimately used by single family homebuilders in the construction of new residential housing. Therefore, the initial working hypothesis is that it appears reasonable that an analysis might find that Woodco s revenues, (Continued on Page 6)

6 earnings and value are significantly influenced by housing trends. Given this working hypothesis, the valuator sets out to objectively test these relationships using regression analysis. In using regression analysis, the goal here is to determine if annual residential housing starts is a statistically significant predictor of a) Woodco annual revenues, and b) Woodco annual earnings (measured by EBITDA), and if so, the direction and nature of those relationships. The following sections outline the results of this analysis. The relationship of Company annual revenues to housing starts is first examined, followed by the relationship of Company profits. SUMMARY OUTPUT Relationship Between Woodco Revenues and Residential Housing Starts ( ) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 18 ANOVA Year Housing Starts (x) Revenues (\$000s)(y) ,146,300 \$12, ,081,400 \$14, ,003,400 \$14, ,900 \$15, ,400 \$14, ,030,100 \$16, ,125,600 \$21, ,198,400 \$26, ,076,300 \$26, ,161,000 \$30, ,133,600 \$29, ,271,400 \$33, ,302,500 \$45, ,230,900 \$50, ,273,200 \$47, ,358,500 \$63, ,499,000 \$69, ,610,500 \$83,400 Relationship of Housing Starts and Company Revenuesb. Relationship of Housing Starts and Company Revenues Regression analysis results are shown below related to the degree to which United States residential housing starts in a given year (the independent variable, x) help predict Woodco s annual revenues (the dependent variable, y): As is shown above, housing starts generally increased materially over the 1989 to 2006 time frame as did Woodco s annual revenues. The statistics shown 6 above enable the analyst to explore if a statistical relationship might exist between the two or whether or not the results are by chance and random. Using the same procedure as before in Excel, a regression of the data, with y (annual revenues) as the dependent variable and x (housing starts) as the independent (and here passive) variable yielding the following results: df SS MS F Significance F Regression E-07 Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Intercept E Housing Starts (x) E Several preliminary conclusions can be drawn from the previous table: Relationship Between Woodco Annual Revenues and Housing Starts is Statistically Significant- Since the F statistic is less than 0.05 this indicates there is a statistically significant relationship present. This indicates that there is a 95% confidence that the results seen would not come up randomly by chance, i.e., that there is a statistically significant relationship present. Strong Relationship of Company Revenues to Housing Starts, With 82.8% of Variations in Woodco Revenues Explained by Changes in National Homebuilding Activity- Given that the regression is valid, the R squared statistic (called the coefficient of determination) indicates that a significant 82.8% of the variations observed in Woodco s annual revenues can be explained by changes in housing starts, a passive force over which Woodco has no control. Positive Relationship Between Woodco Earnings and Housing Starts- The positive nature of the slope coefficient (b) indicates that there is a positive relationship between changes in housing starts and Woodco revenues. This is what might be expected, i.e., as housing starts increase the demand for (Continued on Page 7)

8 A Note to Readers: By necessity, this article used a highly simplified example of regression analysis in business valuation, such as with active passive appreciation issues in divorce, etc. In an actual use of these techniques in a valuation assignment, other statistical techniques might also be used to test the results and larger data sets might be needed. George Hawkins, ASA, CFA Managing Director Banister Financial, Inc. Charlotte, NC Phone: (704) website:

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