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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, 3rd 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 2nd 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, 3rd edition by Roger Gibson Valuing a Business, 4th 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 Bogotá Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto abc McGraw-Hill Copyright © 2001 by McGraw-Hill. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-138595-X The material in this eBook also appears in the print version of this title: 0-07-000215-0. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at [email protected] or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS”. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, 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 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. 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 vii Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. when g ⫽ 0: The Ordinary Annuity. Special Case when n → ⬁ and r ⬎ 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 ⫽ 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 ⬍ 0, Negative Growth, and r ⬍ g. Special Case: No Growth, g ⫽ 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. Appendix A: Automating Iteration Using Newton’s Method. Appendix B: Mathematical Appendix. Appendix C: 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 Introduction. Equity Valuation Method. Table 6-1A: The First Iteration. Table 6-1B: Subsequent Iterations of the First Scenario. Table viii Contents 179 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: Contents ix Reconciliation to the IBA Database. Part 1: IBA P/CF Multiples. Part 2: Log Size P/CF Multiples. Conclusion. Calculation of DLOM. Table 10-4: Computation of the Delay-to-Sale Component–$25,000 Firm. Table 10-5: Calculation of Transactions Costs. Table 10-6: Calculation of DLOM. Table 10-6A–10-6F: Calculations of DLOM for Larger Firms. Calculation of DLOM for Large Firms. Interpretation of the Error. 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 Effects of Absolute Errors in Forecastng Year 1 Cash Flow. Relative Effects of Absolute Errors in Forecasting Year 1 Cash Flow. Absolute and Relative Effects of Relative Errors in Forecasting Year 1 Cash Flow. Absolute Errors in Forecasting Growth and the Discount Rate. Table 11-5: Summary of Effects of Valuation Errors. Summary and Conclusions. PART V SPECIAL TOPICS 12. Valuing Startups 407 Issues Unique to Startups. Organization of the Chapter. First Chicago Approach. Discounting Cash Flow Is Preferable to Net Income. Capital Structure Changes. Venture Capital Rates of Return. Table 12-1: Example of the First Chicago Approach. Advantages of the First Chicago Approach. Discounts for Lack of Marketability and Control. Venture Capital Valuation Approach. Venture Capital Rates of Return. Summary of the VC Approach. Debt Restructuring Study. Backgound. Key Events. Decision Trees and Spreadsheet Calculations. Table 12-3: Statistical Calculation of FMV. Conclusion. Exponentially Declining Sales Growth Model. 13. ESOPs: Measuring and Apportioning Dilution Introduction. What Can Be Skipped. Definitions of Dilution. Dilution to the ESOP (Type 1 Dilution). Dilution to the Selling Owner (Type 2 Dilution). Defining Terms. Table 13-1: Calculation of Lifetime ESOP Costs. The Direct Approach. FMV Equations—All Dilution to the ESOP (Type 1 Dilution; No Type 2 Dilution). Table 13-2, Sections 1 and 2: Post-transaction FMV with All Dilution to the ESOP. The Posttransaction Value Is a Parabola. FMV Equations—All Dilution to the Owner (Type 2 Dilution). Table 13-2, Section 3: FMV Calculations—All Dilution to the Seller. Sharing the Dilution. Equation to Calculate Type 2 Dilution. Tables 13-3 and 13-3A: Adjusting Dilution to Desired Levels. Table 13-3B: Summary of Dilution Tradeoffs. The Iterative Approach. Iteration #1. Iteration #2. Iteration #3. Iteration #n. Summary. Advantages of Results. Function of ESOP Loan. Common Sense Is Required. To Whom Should the Dilution Belong? Appendix A: x Contents 433 Mathematical Appendix. Appendix B: Shorter Version of Chapter 13. 14. Buyouts of Partners and Shareholders 471 Introduction. An Example of a Buyout. The Solution. Evaluating the Benchmarks. Glossary 475 Index 479 Contents xi This page intentionally left blank. Introduction NATURE OF THE BOOK This is an advanced book in the science and art of valuing 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 matters, investment bankers, venture capitalists, financial analysts, and MBA students. Uniqueness of This Book This is a rigorous book, and it is not easy reading. However, the following unique attributes of this book make reading it worth the effort: 1. It emphasizes regression analysis of empirical data. Chapter 7, adjusting for control and marketability, contains the first regression analysis of the data related to restricted stock discounts. Chapter 9, a sample fractional interest discount study, contains a regression analysis of the Partnership Profiles database related to secondary limited partnership market trades. In both cases we found very significant results. We now know much of what drives (a) restricted stock discounts and (b) discounts from net asset values of the publicly registered/ privately traded limited partnerships. You will also see much empirical work in Chapter 4, ‘‘Discount Rates as a Function 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 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. Chapter 4 contains the ADF with constant growth not included in the Mercer text. ADFs crop up in 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 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 to make it easier to follow. Chapters 4 and 13 both contain a shortcut version of the chapter at the end for those who want the bottom line without all the detail. In general, I have moved most of the heaviest mathematics to appendices in order to leave the bodies of the chapters more readable. Where that was not optimal, I have given instructions on which material can 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 value, and (3) adjust for 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 the upcoming material in greater depth than we cover here. Because most professionals do not have abundant time, I want to suggest another path geared for the maximum benefit from the least investment in time. The heart of the book is Chapters 4 and 7, on log size and on adjusting for control and marketability, respectively. I recommend the time-pressed reader follow this order: 1. Chapter 7 (discounts for lack of control and lack of marketability) 2. Chapter 8 (this is an application of Chapter 7—a sample restricted stock report) 3. Chapter 9 (this is an application of Chapter 7—a sample fractional ownership interest discount report) xiv Introduction 4. Chapter 4 (the log size model for calculating discount rates) 5. Chapter 3—the following sections: from the beginning through the section titled ‘‘A Brief Summary’’; ‘‘Periodic Perpetuity Factors: Perpetuities for Periodic Cash Flows’’; and ‘‘Relationship of Gordon Model Multiple to the Price/Earnings Ratio.’’ Some readers may want to read this chapter after Chapter 7, though it will be somewhat helpful, but definitely not necessary, to have read Chapter 3 before 4 and 7. 6. Chapter 10 (this empirically tests Chapters 4 and 7, the heart of the book) 7. Chapter 2 (some readers may want to start with Chapter 2 first, as the material on using regression analysis may help reading all of the other chapters). After these chapters, you can read the remainder of the book in any order, though it is best to read Chapter 14 immediately after Chapter 13. This book has close to 125 tables, many of them being two or three pages long. To facilitate your reading, it will help you to copy tables whose commentary in the text is extensive and sit with the separate tables next to you. Otherwise, you will spend an inordinate amount of time flipping pages back and forth. Note: readers with low blood pressure may wish to ignore that advice. BIBLIOGRAPHY Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts: Developing and Supporting Marketability Discounts in the Appraisal of Closely Held Business Interests. Memphis, Tenn.: Peabody. Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 3d ed. New York: McGraw-Hill. Introduction xv This page intentionally left blank. Acknowledgments I gratefully acknowledge help beyond the call of duty from my parents, Leonard and Marilyn Abrams. Professionally, R. K. Hiatt has been the ideal internal editor. Without his help, this book would have suffered greatly. He also contributed important insights throughout the book. Robert Reilly edited the original manuscript cover-to-cover. I thank Robert very much for the huge time commitment for someone else’s book. Larry Kasper gave me a surprise detailed edit of the first eight chapters. I benefited much from his input and thank him profusely. Chris Mercer also read much or all of the book and gave me many corrections and very useful feedback. I thank Chris very much for his valuable time, of which he gave me much. Michael Bolotsky and Eric Nath were very helpful to me in editing my summary of their work. I thank Rob Oliver and Roy Meyers of Management Planning, Inc. for providing me with their restricted stock data. I also thank Bob Jones of Jones, Roach & Caringella for providing me with private fractional interest sales of real estate. Chaim Borevitz provided important help on Chapters 8 and 9. Mark Shayne provided me with dozens of insightful comments. Professor William Megginson gave me considerable feedback on Chapter 7. I thank him for his wisdom, patience, and good humor. His colleague, Professor Lance Nail, also was very helpful to me. I also appreciate the following people who gave me good feedback on individual chapters (or their predecessor articles): Don Wisehart, Betsy Cotter, Robert Wietzke, Abdul Walji, Jim Plummer, Mike Annin, Ed Murray, Greg Gilbert, Jared Kaplan, Esq., Robert Gross, Raymond Miles, and Steven Stamp. I thank the following people who provided me with useful information that appears in the book: John Watson, Jr., Esq., David Boatwright, Esq., Douglas Obenshain, and Gordon Gregory. I thank the following people who reviewed this book for McGrawHill: Shannon Pratt, Robert Reilly, Jay Fishman, Larry Kasper, Bob Grossman, Terry Isom, Herb Spiro, Don Shannon, Chris Mercer, Dave Bishop, Jim Rigby, and Kent Osborne. xvii Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. I thank my editor at McGraw-Hill, Ela Aktay, for her encouragement and enthusiasm. I thank my publisher, Jeff Krames, for his patience and apologize for testing it so much due to my passion for perfection and the huge life-changes that occurred to me while writing this book. All the people who have helped make this book a reality have my profound gratitude. In fact, there have been so many that it is almost impossible to remember every single person, and I apologize to anyone who should be in this acknowledgment section and who is not due to my human failings. xviii Acknowledgments List of Figures 2-1 2-2 3-1 A3-1 A3-2 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 7-1 7-2 7-3 10-1 12-1 12-2 12-3 12-3A 13-1 Z-distribution vs t-distribution t-distribution of B around the Estimate b Timeline of the ADF and Gordon Model Timeline of Cash Flows Payment Schedule 1926–1998 Arithmetic Mean Returns as a Function of Standard Deviation 1926–1998 Arithmetic Mean Returns as a Function of Ln(FMV) Decade Standard Deviation of Returns versus Decade Average FMV per Company on NYSE 1935–1995 Decade Standard Deviation of Returns versus Decade Average FMV per Company on NYSE 1945–1995 Average Returns Each Decade The Natural Logarithm Discount Rates as a Function of FMV 1939–1998 Decile Standard Deviations as a Function of Ln(FMV) Traditional Levels of Value Chart Two-Tiered Levels of Value Chart 3 ⫻ 2 Levels of Value Chart P/E Ratio as a Function of Size (From the IBA Database) Decision Tree for Venture Capital Funding Decision Tree for Bootstrapping Assuming Debt Restructure and No Venture Capital Sales Forecast (Decay Rate ⫽ 0.5) Sales Forecast (Decay Rate ⫽ 0.3) Post-Transaction Value of the ESOP Vs. % Sold 34 35 65 91 100 125 127 128 129 130 137 137 138 197 198 230 363 421 425 430 431 442 xix Copyright 2001 The McGraw-Hill Companies, Inc. 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