Quantitative Business Valuation Other Titles in the Irwin Library of Investment and Finance Convertible Securities by John P. Calamos Pricing and Managing Exotic and Hybrid Options by Vineer Bhansali Risk Management and Financial Derivatives by Satyajit Das Valuing Intangible Assets by Robert F. Reilly and Robert P. Schweihs Managing Financial Risk by Charles W. Smithson High-Yield Bonds by Theodore Barnhill, William Maxwell, and Mark Shenkman Valuing Small Business and Professional Practices, 3 rd edition by Shannon Pratt, Robert F. Reilly, and Robert P. Schweihs Implementing Credit Derivatives by Israel Nelken The Handbook of Credit Derivatives by Jack Clark Francis, Joyce Frost, and J. Gregg Whittaker The Handbook of Advanced Business Valuation by Robert F. Reilly and Robert P. Schweihs Global Investment Risk Management by Ezra Zask Active Portfolio Management 2 nd edition by Richard Grinold and Ronald Kahn The Hedge Fund Handbook by Stefano Lavinio Pricing, Hedging, and Trading Exotic Options by Israel Nelken Equity Management by Bruce Jacobs and Kenneth Levy Asset Allocation, 3 rd edition by Roger Gibson Valuing a Business, 4 th edition by Shannon P. Pratt, Robert F. Reilly, and Robert Schweihs The Relative Strength Index Advantage by Andrew Cardwell and John Hayden Quantitative Business Valuation A Mathematical Approach for Today’s Professional JAY B. ABRAMS, ASA, CPA, MBA McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogota´ Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Copyright © 2001 by McGraw-Hill. All rights reserved. Manufactured in the United States of America. 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McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURA- CY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/007138595X abc McGraw-Hill To my father, Leonard Abrams, who taught me how to write. To my mother, Marilyn Abrams, who taught me mathematics. To my wife, Cindy, who believes in me. To my children, Yonatan, Binyamin, Miriam, and Nechamah Leah, who gave up countless Sundays with Abba (Dad) for this book. To my youngest child, Rivkah Sarah, who wasn’t yet on the outside to miss the Sundays with me, but who has brought us peace. To my parents and my brother, Mark, for their tremendous support under difficult circumstances. To my great teachers, Mr. Oshima and Christopher Hunt, who brought me to my power to make this happen. And finally, to R. K. Hiatt, who has caught my mistakes and made significant contributions to the thought that permeates this book. This page intentionally left blank. vii Contents Introduction xiii Acknowledgments xvii List of Figures xix List of Tables xxi PART I FORECASTING CASH FLOWS 1. Cash Flow: A Mathematical Derivation 3 Introduction. The Mathematical Model. A Preliminary Explanation of Cash Flows. Analyzing Property, Plant, and Equipment Transactions. An Explanation of Cash Flows with More Detail for Equity Transactions. Considering the Components of Required Working Capital. Adjusting for Required Cash. Comparison to Other Cash Flow Definitions. Conclusion. 2. Using Regression Analysis 21 Introduction. Forecasting Costs and Expenses. Adjustments to Expenses. Table 2-1A: Calculating Adjusted Costs and Expenses. Performing Regression Analysis. Use of Regression Statistics to Test the Robustness of the Relationship. Standard Error of the y Estimate. The Mean of a and b. The Variance of a and b. Selecting the Data Set and Regression Equation. Problems with Using Regression Analysis for Forecasting Costs. Insufficient Data. Substantial Changes in Competition or Product/Service. Using Regression Analysis to Forecast Sales. Spreadsheet Procedures to Perform Regression. Examining the Regression Statistics. Adding Industry-Specific Independent Variables. Try All Combinations of Potential Independent Variables. Application of Regression Analysis to the Guideline Company Method. Table 2-5: Regression Analysis of Guideline Companies. Summary. Appendix: The ANOVA table. 3. Annuity Discount Factors and the Gordon Model 57 Introduction. Definitions. Denoting Time. ADF with End-of-Year Cash Flows. Behavior of the ADF with Growth. Special Case of ADF Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. viii Contents when g ϭ 0: The Ordinary Annuity. Special Case when n → ϱ and r Ͼ g: The Gordon Model. Intuitively Understanding Equations (3-6) and (3-6a). Relationship between the ADF and the Gordon Model. Table 3-1: Proof of ADF Equations (3-6) through (3-6b). A Brief Summary. Midyear Cash Flows. Table 3-2: Example of Equations (3-10) through (3-10b). Special Cases for Midyear Cash Flows: No Growth, g ϭ 0. Gordon Model. Starting Periods Other than Year 1. End-of-Year Formulas. Valuation Date  0. Table 3-3: Example of Equation (3-11). Tables 3-4 through 3-6: Variations of Table 3-3 with S Ͻ 0, Negative Growth, and r Ͻ g. Special Case: No Growth, g ϭ 0. Generalized Gordon Model. Midyear Formula. Periodic Perpetuity Factors (PPFs): Perpetuities for Periodic Cash Flows. The Mathematical Formulas. Tables 3-7 and 3-8: Examples of Equations (3-18) and (3-19). Other Starting Years. New versus Used Equipment Decisions. ADFs in Loan Mathematics. Calculating Loan Payments. Present Value of a Loan. Relationship of the Gordon Model to the Price/Earnings Ratio. Definitions. Mathematical Derivation. Conclusions. PART II CALCULATING DISCOUNT RATES 4. Discount Rates as a Function of Log Size 117 Prior Research. Table 4-1: Analysis of Historical Stock Returns. Regression #1: Return versus Standard Deviation of Returns. Regression #2: Return versus Log Size. Regression #3: Return versus Beta. Market Performance. Which Data to Choose? Recalculation of the Log Size Model Based on 60 Years. Application of the Log Size Model. Discount Rates Based on the Log Size Model. Practical Illustration of the Log Size Model: Discounted Cash Flow Valuations. Total Return versus Equity Premium. Adjustments to the Discount Rate. Discounted Cash Flow or Net Income? Discussion of Models and Size Effects. CAPM. The Fama–French Cost of Equity Model. Log Size Models. Heteroscedasticity. Industry Effects. Satisfying Revenue Ruling 59-60 without a Guideline Public Company Method. Summary and Conclusions. AppendixA: Automating Iteration Using Newton’s Method. AppendixB: Mathematical Appendix. AppendixC: Abbreviated Review and Use. 5. Arithmetic versus Geometric Means: Empirical Evidence and Theoretical Issues 169 Introduction. Theoretical Superiority of Arithmetic Mean. Table 5-1: Comparison of Two Stock Portfolios. Empirical Evidence of the Superiority of the Arithmetic Mean. Table 5-2: Regressions of Geometric and Arthmetic Returns for 1927–1997. Table 5-3: Regressions of Geometric Returns for 1938–1997. The Size Effect on the Arithmetic versus Geometric Means. Table 5-4: Log Size Comparison of Discount Rates and Gordon Model Multiples Using AM versus GM. Indro and Lee Article. 6. An Iterative Valuation Approach 179 Introduction. Equity Valuation Method. Table 6-1A: The First Iteration. Table 6-1B: Subsequent Iterations of the First Scenario. Table Contents ix 6-1C: Initial Choice of Equity Doesn’t Matter. Convergence of the Equity Valuation Method. Invested Capital Approach. Table 6-2A: Iterations Beginning with Book Equity. Table 6-2B: Initial Choice of Equity Doesn’t Matter. Convergence of the Invested Capital Approach. Log Size. Summary. Bibliography. PART III ADJUSTING FOR CONTROL AND MARKETABILITY 7. Adjusting for Levels of Control and Marketability 195 Introduction. The Value of Control and Adjusting for Level of Control. Prior Research—Qualitative Professional. Prior Research— Academic. My Synthesis and Analysis. Discount for Lack of Marketability (DLOM). Mercer’s Quantitative Marketability Discount Model. Kasper’s BAS Model. Restricted Stock Discounts. Abrams’ Economic Components Model. Mercer’s Rebuttal. Conclusion. Mathematical Appendix. 8. Sample Restricted Stock Discount Study 293 Introduction. Background. Stock Ownership. Purpose of the Appraisal. No Economic Outlook Section. Sources of Data. Valuation. Commentary to Table 8-1: Regression Analysis of Management Planning Data. Commentary to Table 8-1A: Revenue and Earnings Stability. Commentary to Table 8-1B: Price Stability. Valuation Using Options Pricing Theory. Conclusion of Discount for Lack of Marketability. Assumptions and Limiting Conditions. Appraiser’s Qualifications. 9. Sample Appraisal Report 315 Introduction. Purpose of the Report. Valuation of Considerations. Sources of Data. History and Description of the LLC. Significant Terms and Legal Issues. Conclusion. Economic Outlook. Economic Growth. Inflation. Interest Rates. State and Local Economics. Summary. Financial Review. Commentary to Table 9-2: FMV Balance Sheets. Commentary to Table 9-3: Income Statements. Commentary to Table 9-4: Cash Distributions. Valuation. Valuation Approaches. Selection of Valuation Approach. Economic Components Approach. Commentary to Table 9-5: Calculations of Combined Discounts. Commentary to Table 9-5A: Delay-to-Sale. Commentary to Table 9-5C: Calculation of DLOM. Commentary to Table 9-6: Partnership Profiles Approach—1999. Commentary to Table 9-7: Private Fractional Interest Sales. Commentary to Table 9-8: Final Calculation of Fractional Interest Discounts. Conclusion. Statement of Limiting Conditions. Appraiser’s Qualifications. Appendix: Tax Court’s Opinion for Discount for Lack of Marketability. Introduction. The Court’s 10 Factors. Application of the Court’s 10 Factors to the Valuation. PART IV PUTTING IT ALL TOGETHER 10. Empirical Testing of Abrams’ Valuation Theory 357 Introduction. Steps in the Valuation Process. Applying a Valuation Model to the Steps. Table 10-1: Log Size for 1938–1986. Table 10-2: [...]... Conclusion 11 Measuring Valuation Uncertainty and Error 383 Introduction Differences Between Uncertainty and Error Sources of Uncertainty and Error Measuring Valuation Uncertainty Table 11-1: 95% Confidence Intervals Summary of Valuation Implications of Statistical Uncertainty in the Discount Rate Measuring the Effects of Valuation Error Defining Absolute and Relative Error The Valuation Model Dollar... of Log Size,’’ and Chapter 11, ‘‘Empirical Testing of Abrams’ Valuation Theory.’’ 2 It emphasizes quantitative skills Chapter 2 focuses on using regression analysis in business valuation Chapter 3, ‘‘Annuity Discount Factors and the Gordon Model,’’ is the most comprehensive treatment of ADFs in print For anyone wishing to use the Mercer quantitative marketability discount model, xiii Copyright 2001... privately held businesses In order to read this book, you must already have read at least one introductory book such as Valuing A Business (Pratt, Reilly, and Schweihs 1996) Without such a background, you will be lost I have written this book with the professional business appraiser as my primary intended audience, though I think this book is also appropriate for attorneys who are very experienced in valuation. .. many valuation contexts I invented several new ADFs that appear in Chapter 3 that are useful in many valuation contexts Chapter 11 contains the first treatise on how much statistical uncertainty we have in our valuations and how value is affected when the appraiser makes various errors 3 It emphasizes putting all the pieces of the puzzle together to present a comprehensive, unified approach to valuation. .. Log Size Model Absolute Errors in Forecasting Growth Rates Percent Valuation Error for 10% Relative Error in Growth Percent Valuation Error for Ϫ10% Relative Error in Growth 374 374 375 375 376 376 377 377 389 391 398 400 401 11-4B 11-5 12-1 12-2 Percent Valuation Error for 10% Relative Error in Discount Rate Summary of Effects of Valuation Errors First Chicago Method VC Pricing Approach 402 403 412... Comparison of Discount Rates Derived from the Log Size Model Using 60-Year Arithmetic and Geometric Means 176 6-1A Equity Valuation Approach with Iterations Beginning with Book Equity: Iteration #1 182 6-1B Equity Valuation Approach with Iterations Beginning with Book Equity 184 6-1C Equity Valuation Approach with Iterations Beginning with Arbitrary Equity 185 6-2A WACC Approach with Iterations Beginning... be safely skipped How to read this book depends on your quantitative skills and how much time you have available For the reader with strong quantitative skills and abundant time, the ideal path is to read the book in its exact order, as there is a logical sequence The first three parts to this book follow the chronological sequence of performing a valuation: (1) forecast cash flows, (2) discount to present... (ADFs) Historically, ADFs have not been used much in business valuation and thus, have had relatively little importance Their importance is growing, however, for several reasons They can be used in: ● ● ● ● ● ● Calculating the present value of annuities, including those with constant growth This application has become far more important since the Mercer Quantitative Marketability Discount Model requires... together to present a comprehensive, unified approach to valuation that can be empirically tested and whose principles work for the valuation of billion-dollar firms and ma and pa firms alike While this book contains more mathematics—a worm’s eye view, if you will—than other valuation texts, it also has more of a bird’s eye view as well HOW TO READ THIS BOOK I have tried to provide paths through this book... control and marketability The fourth part is a bird’s eye view in order to test empirically whether my methodology works Additionally, we explore (1) confidence intervals around valuation estimates and (2) what happens to the valuation when appraisers make mistakes Part 5, on special topics, is the place for everything else Each of parts of the book has an introduction preceding it that will preview . Testing of Abrams’ Valuation Theory.’’ 2. It emphasizes quantitative skills. Chapter 2 focuses on using regression analysis in business valuation. Chapter. Relative Strength Index Advantage by Andrew Cardwell and John Hayden Quantitative Business Valuation A Mathematical Approach for Today’s Professional JAY B.