< PreviousFurther, in contrast with the prevailing value reconciliation practice mechanisms, nearly 75 percent of respondents to a 2020 NACVA survey conducted as part of Salter’s study (2020) agreed that values derived from the three methods would be in sync if the common factors underlying those methods are in sync (Salter 2020, 148–152). When asked if they had a method to enable in-sync common factors, only 13 percent indicated they had a method for adjusting discount rates between economic income streams to enable this (Salter 2020, 118). Of the 13 percent, one respondent noted their use of Damodaran’s approach referenced above, another referenced the use of “triangulation,” some referenced adjusting discount rates proportionately to different economic income stream measures, and others were more vague. Regardless of an abundance of literature supporting a mathematical mechanism to reconcile the common factors, other than the above references, the literature is generally silent on a comprehensive practice mechanism that can be broadly employed to facilitate in-sync common factors to reconcile going concern value indications between the three methods. Table 2: Potential Practice Mechanisms YearScholarInformation 2008 Pratt and Niculita Regarding values from the three methods applied to single economic income streams, Pratt and Niculita stated that “with the same set of assumptions, the discounted economic income method and the capitalized economic income method using the Gordon growth model with a 5 percent growth rate will produce an identical valuation indication. Therefore, an analyst using the capitalized economic income method should understand its parent method (the discounted economic income method) and think through, as a form of mental verification of reasonableness, ‘If I carried out the full discounting procedure, would I get approximately the same answer?’ If not, the valuation variables used in the capitalization method should be reexamined, or perhaps the capitalization method should only be used for a terminal value in conjunction with the discounting method." (p. 245) 2008 Pratt and Niculita Regarding values from the three methods applied to multiple economic income streams, Pratt and Niculita stated that any economic income stream can be “converted into a value indication through the use of a discount (or capitalization) rate appropriate to the measure of economic income. Of course, different discount (or capitalization) rates would be appropriate to the different measures of economic income.” (p. 175) 2012Damodaran Regarding reconciling value from the three methods applied to individual or multiple economic income streams, Damodaran stated: “Some analysts who use multiples go back to these discounted cash flow models to extract multiples. Other analysts compare multiples across firms or time and make explicit or implicit assumptions about how firms are similar or vary on fundamentals.” (p. 20) A fundamentals “approach relates multiples to fundamentals about the firm being valued—growth rates in earnings and cash flows, reinvestment and risk. This approach to estimating multiples is equivalent to using discounted cash flow models, requiring the same information and yielding the same result. Its primary advantage is that it shows the relationship between multiples and firm characteristics, and allows us to explore how multiples change as these characteristics change.” (p. 20) 2015 Chastenet and Marion “According to the theoretical model of discounted free cash flows, the multiples method assumes that it is possible to identify publicly traded companies, said to be comparable, in that they have a profile of free cash flows, growth prospects, and a level of risk similar to those of the company that is being valued. Considering the single-period model derived from the Gordon and Shapiro (1956) formula, a simplification of the discounted free cash-flows method, any synthetic valuation estimated multiple (Mest) derived from a sample of comparable companies can be assimilated to a capitalization factor applied to the expected free cash flow (CF) of the company to be valued.” (p. 174) 10The Value Examiner ValuationEnhanced Interpretation of the Scholarly and Practitioner Literature: Reconciling the Separate Value Hierarchy Levels As noted, the second area of value reconciliation involves something not typically thought of as needing reconciliation: potential differences in going concern value between different value hierarchy levels. The literature addressing this topic includes (1) literature identifying the value hierarchy itself along with the puzzle of its seemingly contradicting theories, and (2) literature describing the primary way that scholars and practitioners have indirectly considered the topic. Classic valuation theories initially identifying the value hierarchy. WGS introduced dividend-oriented theories as a rebuke to the speculative price-to-earnings (P/E) multiple-oriented pre-1929 investor behavior precipitating the Great Depression (Salter 2020, 18–20). Many studies, however, demonstrate that WGS’s theories do not fully explain market values for most publicly traded businesses (Salter 2020, 18–38). MM augmented WGS’s theories with their acquisition- oriented theories (1958, 1961), arguing that business value is not limited only to the companies forecasting dividends, given a company’s potential value to financial and strategic acquirers (Salter 2020, 25–31). How scholastic and practitioner literature addresses the value hierarchy. Generally, the literature addresses the reconciliation of going concern value hierarchy levels by ascribing premia and discounts between them—i.e., MM financial control value over WGS non-control dividend value (if there is a difference as noted by Nath in 2011 and 2013), and MM strategic control over MM financial control value. These subjects are widely covered in professional practitioner valuation practice standards and textbooks,13 as well as empirical studies by practitioners and other scholars.14 While the prevailing practice methods prompt professional practitioners to use informed judgment to ascribe premia and discounts based on the comparison of the situational characteristics of any going concern against premia and discounts in various databases, such approaches do not represent a reconciliation of the three possible value hierarchy levels themselves. In effect, once going concern FMV is determined, premia or discounts can be applied, but there is no practice mechanism that can be used to reconcile three possible value hierarchy levels to determine FMV in the first place. 13 See Pratt and Niculita (2008), Damodaran (2012), Trugman (2012), and others. 14 For example, Pratt and Niculita (2008, 384, 386); Trevino (2009, 140–143); Nath (2011, 2013); Covrig, McConaughy, and Travers (2016). The Alternative Practice Mechanism The process of reconciling the going concern value indications between the three methods and value hierarchy levels is the focus of Salter’s APM. The APM provides an ordered process for valuators to arrive at conclusions possessing greater support than prevailing practice mechanisms. There are two parts to the APM: (1) a practice mechanism that reconciles the separate value hierarchy measures, called the augmented valuation theory (AVT), facilitated through an equation called the AVT theorem, and (2) a multistep process facilitating in- sync value indications between the three methods, called the in-sync methodology. Augmented Valuation Theory and the AVT Theorem The addition of MM’s theories to those of WGS confirms that, other than liquidating a business’s assets, there are two ways going concerns can yield value to equity holders: cash payments to equity holders (e.g., dividends or distributable FCFE) or proceeds from a going concern sale. There are no other options. On this topic, a majority of respondents to Salter’s NACVA survey confirmed that MM strategic-acquirer-oriented measures equal or exceed MM financial acquirer levels, and MM financial acquirer levels equal or exceed WGS shareholder dividend- oriented levels. As such, each of these value hierarchy levels are mutually exclusive of one another, and together provide a collectively exhaustive way that going concerns can possess value (Salter 2020, 32–36, 81–83, 122–125). In short, the three possible value hierarchy measures represent mutually exclusive and collectively exhaustive ways going concerns can yield value to equity holders other than from a liquidation of assets. Further, as MM’s acquirer-oriented theories depend on an acquisition, it is necessary to ask whether the discount rates underlying the three methods capture the probability of an acquisition being completed (Salter 2020, 126–131). While marketability discounts within the literature address the salability of minority interests (Salter 2020, 131–133), little in the literature addresses the topic of overall acquisition probability. When asked in Salter’s NACVA survey, nearly two-thirds of respondents agreed that the capital asset pricing model (CAPM)-derived cost of equity (Ke) does not include acquisition probability, and more than 80 percent agreed that the probability of acquisition needs to be factored into final valuation conclusions (Salter 2020, 129). 11March | April 2022 A Professional Development Journal for the Consulting DisciplinesGiven the above, WGS’s and MM’s valuation theories need to be augmented to reflect their mutually exclusive and collectively exhaustive nature and the need to assess the probability of an acquisition. Utilizing something called the MECE principle (Minto 1996)15 and the multiplication rule of probability16 (Salter 2020, 6, 34), going concern FMV can best be understood as a composite of WGS shareholder dividend-oriented value plus some probability-based acquisition value (if there is any). In many ways, going concern FMV measures represent a form of base-WGS- value plus MM acquisition-option value. More fulsomely, and as portrayed in Figure 2, going concern FMV can be understood as being equal to: 1. The probability that a strategic acquirer (Ps in Figure 2, Scenario 1) bids on and completes an acquisition of a going concern at an MM strategic acquirer-oriented going concern business value (S in Figure 2); plus 2. The probability that a strategic buyer does not bid on and complete an acquisition ( 1-Ps in Scenario 2), along with the probability that a financial acquirer does bid on and complete an acquisition ( Pf in Scenario 2) at an MM financial acquirer-oriented going concern business value ( F in Scenario 2); plus 3. The probability that neither a financial nor strategic acquirer bid on and complete an acquisition—( 1-Ps) * ( 1-Pf) in Scenario 3—leaving going concern FMV to equal the WGS shareholder dividend-oriented value (noting that such FMV would always be compared against a liquidation approach that goes beyond the scope of this article; Salter 2020, 5–6, 32–36). Figure 2: The AVT Theorem 15 The mutually exclusive and collectively exhaustive (MECE) principle was developed by Barbara Minto in the late 1960s and published in the U.S. in 1987 (Minto 1996). 16 The multiplication rule of probability states that in the event of multiple possible outcomes arising in a single event or situation (e.g., if multiple outcomes—such as A, B, and C—have a possibility of arising), the probability that A, B, and C occur is equal to the probability that A occurs multiplied by the probability that B occurs multiplied by the probability that C occurs. 17 Thanks to Salter’s colleague, Eric Briggs, for associating Salter’s theories with the stated mathematical principles and rules. Regardless of whether the AVT theorem is applied as an actual practice mechanism, or just used as a guide to address the topics embodied within it, it resolves the long-standing issue of value hierarchy reconciliation and provides a useful guide to enable scholars and professional practitioners to better focus informed judgment in both research and analytic efforts.17 In-Sync Methodology Regarding the first of the two going concern valuation reconciliation topics, Salter expounded on the guidance provided by Damodaran (2012, 20), Pratt and Niculita (2008, 175), and Chastenet and Marion (2015, 174) by developing a mechanism supported by the literature and research that reconciles going concern valuation indications derived from the three methods as applied to single and multiple economic income streams. Salter’s in-sync methodology facilitates this goal in a two-stage multi-step process (Salter 2020, 155–178). In the first stage, preliminary valuation indications are developed at each value hierarchy level with in-sync common factors. In the second stage, those preliminary common factors are compared to those of a group of guideline public companies (GPCs) to finalize the valuation indications at each value hierarchy level. Thereafter, the going concern valuation is finalized by applying the AVT theorem as described above. Together, the in-sync methodology and AVT theorem make up the APM (Salter 2020). First-stage in-sync methodology steps: preliminary going concern values and common factors. The first step is to determine preliminary equity, debt, and enterprise value for a going concern using the discounted forecast method applied against effective perpetuity dividends or distributable FCFE for equity, and cash-flow-to-debt for debt at each value hierarchy level. To distinguish between value hierarchy levels, consideration is first given to whether the “as is” business is operating and accessing capital at a level like MM financial acquirers and, if not, whether MM financial- acquirer adjustments might be applied. Consideration is next given to possible revenue and cost-saving synergies along with capital market options (Salter 2020, 155–162). The next step is to determine initial implied discount rates applicable to the other economic income streams by calculating the internal rates of return at each economic income stream level from the effective perpetuity economic 12The Value Examiner Valuationincome stream forecasts applied against the preliminary equity and enterprise valuations referenced in the above paragraph (Salter 2020, 51–55). The last step is to determine implied perpetuity growth rates that are explicit in the capitalized growth method and implicit in the market multiple method, noting these growth rates are likely different than the compound growth rate in perpetuity forecasts, given that they are time-value- weighted (Salter 2020, 46–48). Once completed, the valuation levels between the three methods will be reconciled, and the preliminary common factors underlying the three methods at each economic income stream will be in sync within each of the three value hierarchy levels. Second-stage in-sync methodology steps: final going concern values and common factors. As noted, the second-stage in-sync methodology steps aim to finalize the preliminary going concern valuation indications at each value hierarchy level by comparing the going concern’s preliminary common factors at each economic income stream with those of the GPCs. Given the in-sync methodology’s first stage producing going concern common factors, what is now needed are common factors for the GPCs at each economic income stream. The GPC information typically available to analysts includes market multiples at each economic income stream, and CAPM (or equivalent) calculated Ke’s, estimated Kd’s, and implied perpetuity growth rates, which can be factored from GPC P/E multiples using the formula g = ((Ke x PE)-1))/(PE + 1), where g = growth, Ke = cost of equity, and PE = the P/E multiple (Salter 2020, 48–49, 178–179). What is now needed are (1) implied effective perpetuity growth rates to apply to each GPC economic income stream representative level, and (2) implied discount rates that apply to each GPC economic income stream representative level. Regarding the effective perpetuity growth rate at each GPC economic income stream, Salter’s study demonstrated that perpetuity growth rates converge for a range of economic income streams of any hypothetical going concern over time (Salter 2020, 163–164). As such, the above-referenced implied perpetuity growth rate derived from the formula in the above paragraph can be used as a proxy across all GPC economic income streams. Regarding discount rates applicable to each GPC economic income stream, these can now easily be computed using the GPC market multiples and implied effective perpetuity growth rate at each GPC economic income stream level using the formula K = ((1 + g)/market multiple) + g (Salter 2020, 179). With this expanded information, the Ke, Kd, and perpetuity growth rates preliminarily selected for the going concern can either be confirmed or modified to reflect more nuanced comparative investment risk and growth between the going concern and the GPCs, from which going concern valuation indications can be finalized at each value hierarchy level. Upon completion of the in-sync methodology, the potentially different going concern valuation indications at each value hierarchy level can be applied to the AVT theorem, after which the going concern FMV is finalized, with such final number reflecting reconciliation of the three methods as applied within and between economic income streams, and reconciliation of the potentially separate value hierarchy measures. 13March | April 2022 A Professional Development Journal for the Consulting DisciplinesConclusion The APM, composed of the in-sync methodology and AVT as facilitated through the AVT theorem, contributes to our understanding of the valuation of going concerns. The APM’s introduction enhances the scholarly literature by contextualizing each aspect of financial and valuation theory within the separate parts of the APM. The APM provides a basis to foster stronger understanding of issues surrounding going concern FMV, including a way to focus the application of informed judgment around a going concern’s comparative contextual strengths, weaknesses, threats, and opportunities relative to the markets in which it operates. The practical implications of the APM include the potential improvement of broad transactional markets, courtroom processes, and educational processes. Rather than relying on the prevailing “informed judgment” and “averaging” mechanisms that offer limited decision frameworks or support within the literature, differences in going concern FMV opinions can focus on the common factors between the three methods within and between economic income streams at each value hierarchy level. Further, reconciliation of possible differences in the value hierarchies can be more fulsomely debated in respect to (1) possible differences in operating and capital market efficiencies that enable potential value hierarchy differences, and (2) the probability of an acquisition being applied against MM acquirer- oriented value indications. In future articles on the APM, the authors will present examples of the APM process and introduce an augmented version of the APM to address the reconciliation of FMV for “uncertain going concerns,” meaning non-distressed businesses unable to forecast dividends or FCFE into effective perpetuity. Roy Salter, DBA, is a senior advisor at FTI Consulting, LLC, in Los Angeles. He provides forecasting, valuation, financial opinion, and transaction support in a broad range of situations on behalf of a wide range of businesses, with a particular emphasis on the entertainment, media, and broader intellectual property sectors. Dr. Salter drafted a 2020 doctoral dissertation titled “Reconciling Going-Concern Business Valuation Indications Derived from Multiple Business Valuation Methods,” which forms the basis of this article. Dr. Salter is a clinical research professor teaching introduction to entertainment and media finance, has served on several boards, and is active in a broad range of philanthropic initiatives. Vicentiu Covrig, PhD, is a professor of finance at California State University, Northridge. His primary research interests are in business valuation and international finance. He has published in the Journal of Finance, the Journal of Financial and Quantitative Analysis, the Journal of Banking and Finance, the Business Valuation Review, and other publications. He is a member of the academic board of the Business Valuation Review, and recent research has focused on the valuation of complex corporate securities and the determination of the financial metric (i.e., revenues, profits) volatility needed as input to the models. He has made numerous presentations at academic and practitioner conferences, including NACVA and American Society of Appraisers national conferences. Akanksha Bedi, PhD, is an assistant professor at Western Washington University. Her research areas include the study of organizational leadership and investigation of the causes and effects of dysfunctional work behaviors. She studies these phenomena through the analytical lens of meta-analyses. Her research has been published in journals such as the Academy of Management Learning & Education, the Journal of Business Ethics, and Applied Psychology: An International Review. Additionally, Dr. Bedi has worked on various consulting and business projects, including the Youth Policy Institute, the Los Angeles Performance Partnership Pilot (LA P3) Program, the Economic and Workforce Development Department (EWDD), and the Community Action Board. 14The Value Examiner ValuationReferences American Institute of Certified Public Accountants (AICPA). 2015. Statement on Standards for Valuation Services. New York: AICPA. American Society of Appraisers (ASA). 2009 (February 2022 release). ASA Business Valuation Standards. Herndon, VA: ASA. Appraisal Standards Board. 2016. 2016–2017 Uniform Standards of Professional Appraisal Practice. Standard 9, Business Appraisal, Development. Washington, DC: The Appraisal Black, Fischer. 1976. “The dividend puzzle.” Journal of Portfolio Management 2, no. 2 (Winter): 5–8. 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Yoo, Yong Keun. 2006. “The Valuation Accuracy of Equity Valuation Using a Combination of Multiples.” Review of Accounting & Finance 5 (2): 108–123. 15March | April 2022 A Professional Development Journal for the Consulting DisciplinesMuch of the business valuation literature regarding long-term growth rates deals with a firm’s ability to sustain long-term growth, usually in the context of the discounted cash flow (DCF) model. The DCF method of business valuation requires the valuator to forecast growth rates over a discrete number of years. For longer forecasting periods (e.g., over five years), it will likely be necessary to make assumptions based on macroeconomic indicators or historical industry growth rates. The objective of the research discussed in this article is to examine industry growth forecasts based on 10-year inflation forecasts published by the Congressional Budget Office (CBO), 10-year gross domestic product (GDP) forecasts published by the CBO, and 10-year historical average industry growth rates for 21 industries for 2000 through 2010. This research tests the accuracy of the three forecasting methods, measures whether there is a statistically significant difference between them, and determines which of the three methods is superior. Forecasting Growth Rates 1 Shannon P. Pratt and Alina V. Niculita, Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th ed. (New York: McGraw Hill, 2008). 2 Gary R. Trugman, Understanding Business Valuation: A Practical Guide to Valuing Small to Medium Sized Businesses, 3rd ed. (New York: American Institute of Certified Public Accountants, 2008); Greg R. Caruso, The Art of Business Valuation: Accurately Valuing a Small Business (Hoboken, NJ: John Wiley & Sons, 2020). 3 Everett P. Harry III and Jeffrey H. Kinrich, Lost Profits Damages: Principles, Methods, and Applications (Ventnor City, NJ: Valuation Products and Services, 2017); Nancy J. Fannon and Jonathan M. Dunitz, The Comprehensive Guide to Economic Damages, 5th ed. (Portland, OR: Business Valuation Resources, 2018). 4 Roger J. Grabowski, “Comparing Growth Rates Used in Discounted Cash Flow Valuations,” Business Valuation Review 40, no. 1 (Winter 2021): 2–12. The growth rate is a necessary input into business valuation methods under the income approach.1 Long-term growth rates are forecasted in perpetuity; however, this article examines shorter forecast horizons—10 years or less—which are common when using the DCF model or calculating lost profits. The discrete period over which cash flows are projected when using the DCF model is generally between three and 10 years, with five years being the most common in practice.2 In commercial litigation, lost profits are generally calculated for periods less than 10 years. Post-incident damage periods should be reasonably certain and credible.3 A lost profit forecast beyond 10 years will likely be met with skepticism. When projecting growth rates, valuation analysts have few options. They include: firm-specific analysis, econometrics, macroeconomic trends, and historical industry average growth rates. The long-term growth rate has been discussed in the business valuation literature from the perspective of the subject firm’s ability to reinvest cash flows.4 Firm-specific By Gene A. Trevino, PhD, CFA, ASA Forecasting Growth: Macroeconomic Indicators versus Historical Averages 16The Value Examiner Valuationfinancial analysis is paramount to a comprehensive business valuation, but it is questionable whether such analysis can accurately forecast growth rates over a 10-year horizon. Regression analysis is a popular econometric tool for forecasting, but it is valid only for the range of values used in estimating the regression equation, limiting its usefulness for long-term forecasting.5 Given the shortcomings of firm-specific analysis and econometrics, analysts are often relegated to relying on published long-term forecasts of macroeconomic indicators and historical industry growth rates. Reliable guides for long-term growth rates are expected inflation and growth in the overall economy.6 Two relevant macroeconomic indicators for forecasting long-term growth are inflation growth and GDP growth. Logic dictates that inflation is the lower bound because prices increase over time. A firm’s growth, at a minimum, must keep pace with inflation or it will no longer be economically feasible. GDP is the upper bound because no business can consistently grow at a faster rate. One would expect a firm’s long-term growth rate to collapse on an unobservable trendline bounded by inflationary growth and GDP growth. The historical long-term average growth rate in the relevant industry is also a reasonable basis for formulating a long-term growth rate. Despite the subjectivity involved in choosing the lookback period, a cogent argument can be made that future industry-specific uncertainties are “baked into” the historical average industry growth rate. This research tests the accuracy of long-term industry growth rate forecasts premised on 10-year inflation and GDP forecasts published by the CBO and 10-year historical average industry growth rates. It tests the forecasts for accuracy over the period 2010 through 2020, determines whether there are any significant differences between the forecasting methods, and identifies the superior method. Data The industry data used in this research is gross output by industry as reported by the Bureau of Economic Analysis. Gross output by industry is defined as industrywide sales or receipts, and includes sales to consumers and businesses.7 Generally, in the context of projecting discrete cash flows or lost profits, gross sales are forecasted and then reduced by the relevant costs. Industry gross output data was used 5 Gerald J. Hahn, “The Hazards of Extrapolation in Regression Analysis,” Journal of Quality Technology 9, no. 4 (1977): 159–165. 6 Z. Christopher Mercer and Travis W. Harms, Business Valuation: An Integrated Theory, 3rd ed. (Hoboken, NJ: John Wiley & Sons, 2021). 7 “Gross Output by Industry,” Bureau of Economic Analysis, updated January 27, 2022, https://www.bea.gov/data/industries/gross-output-by-industry. 8 For more information regarding the Bureau of Economic Analysis’s industry data, see Bureau of Economic Analysis, Measuring the Nation’s Economy: An Industry Perspective, May 2011, https://www.bea.gov/sites/default/files/methodologies/industry_primer.pdf. 9 Congressional Budget Office, The Budget and Economic Outlook: Fiscal Years 2010 to 2020, January 6, 2010, 10 Congressional Budget Office, CBO’s Forecasting Record: 2021 Update, December 2021, https://www.cbo.gov/system/files/2021-12/57579-forecast-record.pdf. because it can form the basis for projecting a firm’s long- term gross sales.8 The industry data for 2000 through 2010 was used to calculate 10-year historical average growth rates. These historical average growth rates were used as growth rate forecasts for 21 selected industries between 2010 and 2020. The industries examined in this research are: 1. Agriculture—farms 2. Oil and gas extraction 3. Utilities 4. Construction 5. Manufacturing—durable 6. Manufacturing—nondurable 7. Wholesale 8. Retail 9. Transportation—truck 10. Warehousing and storage 11. Information 12. Finance and insurance 13. Real estate, rental, and leasing 14. Professional and business services 15. Professional, scientific, and technical services 16. Management of companies and enterprises 17. Education 18. Healthcare and social assistance 19. Arts, entertainment, and recreation 20. Accommodation 21. Food services and drinking establishments These are broad industry categories, with the exception of numbers 1, 2, 5, 6, 9, and 10. For example, retail trade includes the following sub-industries: motor vehicle and parts dealers, food and beverage stores, general merchandise stores, and other retail. When using this information, one should endeavor to use the industry that best represents the subject firm’s operations. The forecasts for inflation and GDP growth were taken from the CBO’s economic projections for 2010 through 2020.9 The CBO is a credible government source for long-term economic forecasts and publishes the accuracy of its forecasts.10 The GDP forecast was based on nominal GDP and the inflation forecast was based on the Consumer Price Index—All Urban Consumers. 17March | April 2022 A Professional Development Journal for the Consulting DisciplinesMethodology The accuracy of industry growth rate forecasts using CBO inflation forecasts, CBO GDP forecasts, and 10-year historical average industry growth rates was tested by comparing the forecasted growth rates for the period 2010 through 2020 to actual industry growth rates. The forecasted growth rates for inflation and GDP, as reported by the CBO in January 2010, are presented in Table 1. The average industry growth rates used to forecast growth rates from 2010 to 2020, based on 2000 to 2010 historical averages, are presented in Table 2. The forecasted growth rates in Tables 1 and 2 were compared to actual industry growth rates from 2010 through 2020 to test their accuracy. The mean absolute error (MAE) was used to measure the accuracy of forecasted growth rates relative to actual growth rates. The MAE is a method of testing errors between paired actual and predicted values and is an appropriate measure of average forecast errors.11 Simply put, the MAE measures how big an error one can expect on average. It is calculated as the average of the absolute values of the differences between forecasted values and actual values. Absolute values are used to prevent positive and negative differences from cancelling each other out. The MAE can be expressed mathematically as follows: The MAE was calculated for each industry based on the three forecasting methods. By comparing the MAE for individual industries, one can determine which forecasting method is superior. For purposes of identifying the superior method across all industries, the MAEs for each forecasting method were averaged. In order to determine whether there is a statistically significant difference between the average MAEs across all industries an analysis of variance (ANOVA) test was used. ANOVA is a statistical test used to determine if there is a statistically significant difference between population means. The null hypothesis of the ANOVA test is that there is no difference between the means (Ho: u1=u2=u3). 11 Cort J. Willmott and Kenji Matsuura, “Advantages of the Mean Absolute Error (MAE) over the Root Mean Square Error (RMSE) in Assessing Average Model Performance,” Climate Research 30, no. 1 (2005): 79–82. 12 The Tukey-Kramer test is a statistical test used to determine which means differ from each other. The alternative hypothesis is that there is a statistically significant difference between the means (Ha: u1≠u2≠u3). Th e results of this analysis will reveal whether there is a statistically significant difference between the three forecasting methods. If such a difference is found among the average MAEs for the three forecasting methods, a Tukey-Kramer post-hoc test12 will be used to identify which average MAEs differ from each other. Results The MAEs for the three forecasting methods, by industry, are presented in Table 3. The oil and gas extraction industry had the highest MAE (0.2293) premised on the inflation-based forecast. The finance and insurance industry had the lowest MAE (0.0085) premised on the historical average industry growth-based forecast. At the foot of the columns in Table 3, the average MAEs are presented. As one would expect, forecasted growth rates based on the published CBO inflation forecast produced the highest MAEs for most of the industries, as is evident by the relatively higher average MAE (0.0536). Generally, using expected inflation as a forecasting method will underestimate long-term growth for an industry or firm because it does not allow for real growth. Technically, the GDP-based forecasts produced the lowest average MAE (0.0433). That said, there was little difference between the average MAE for the GDP- based forecast (0.0433) and the historical industry average forecasts average MAE (0.0457). Because inflation-based forecasts produced the highest average MAE, they should be eschewed in favor of GDP-based forecasts and historical average growth rate forecasts. The ANOVA results— F (2, 40) = 13.44, P <.0001—revealed that there is a statistically significant difference between the average MAEs for the three forecasting methods. The Tukey-Kramer post-hoc test reveals a significant difference between the average MAE for the inflation- based forecasts and the average MAE for the GDP-based forecasts as evidenced by t (40) = 4.95, P <.0001. There was also a statistically significant difference between the average MAEs for the inflation-based forecasts and the historical industry average-based forecasts, as evidenced Table 1: Forecasted Inflation and GDP Growth Rates 201020112012–2014 2015–2020 Consumer Price Index—All Urban Consumers1.60%1.10%1.30%1.90% Nominal Gross Domestic Product3.20%2.80%5.60%4.20% 18The Value Examiner ValuationTable 2. 2000–2010 Historical Average Industry Growth Rates Growth % Agriculture—farms4.73% Oil and gas extraction16.38% Utilities5.31% Construction2.04% Manufacturing—durable0.47% Manufacturing—nondurable4.25% Wholesale4.90% Retail2.95% Transportation—truck3.11% Warehousing and storage10.03% Information3.08% Finance and insurance4.21% Real estate, rental, and leasing5.02% Professional and business services4.69% Professional, scientific, and technical services4.75% Management of companies and enterprises5.46% Educational services7.70% Healthcare and social assistance6.41% Arts, entertainment, and recreation5.13% Accommodation3.17% Food services and drinking establishments4.39% 19March | April 2022 A Professional Development Journal for the Consulting DisciplinesNext >