Quantitative Business Valuation: A Mathematical Approach for Today's Professionals

Quantitative Business Valuation A Mathematical Approach for Today's Professionals

Essential reading for the serious business appraiser, Quantitative Business Valuation, Second Edition is the definitive guide to quantitative measurements in the valuation process. No other book written on business valuation is as well researched, innovative, and bottom-line beneficial to you as a practitioner.

Written by leading valuation and litigation economist Jay B. Abrams, this text is a rigorous and eye-opening treatment filled with applications for a wide variety of scenarios in the valuation of your privately held business.

Substantially revised for greater clarity and logical flow, the Second Edition includes new coverage of:

• Converting forecast net income to forecast cash flow
• Damages in manufacturing firms
• Regressing scaled y-variables as a way to control for heteroscedasticity
• Mathematical derivation of the Price-to-Sales (PS) ratio
• Monte Carlo Simulation (MCS) and Real Options (RO) Analysis
• Venture capital and angel investor rates of return
• Lost inventory and lost profits damage formulas in litigation

Organized into seven sections, the first three parts of this book follow the chronological sequence of performing a discounted cash flow. The fourth part puts it all together, covering empirical testing of Abrams' valuation theory and measuring valuation uncertainty and error. Parts five to seven round it all out with discussion of litigation, valuing ESOPs and partnership buyouts, and probabilistic methods including valuing start-ups.

The resulting work, solidly grounded in economic theory and including all necessary mathematics, integrates existing science into the valuation profession—and develops valuation formulas and models that you will find useful on a daily basis.

List of Tables and Figures xiii

Introduction xxi

Acknowledgments xxvii

Part I Forecasting Cash Flow 1

Chapter 1 Cash Flow: A Mathematical Derivation 5

Introduction 7

The Mathematical Model 11

Analysis of the Mathematical Model 25

Summary 27

References 27

Chapter 2 Forecasting Cash Flow: Mathematics of the Payout Ratio 29

Introduction 31

The Mathematics 32

Forecasting Gross Cash Flow Is Incorrect 43

Conclusion 44

References 44

Chapter 3 Using Regression Analysis 45

Introduction 47

Forecasting Costs and Expenses 48

Performing Regression Analysis 51

Use of Regression Statistics to Test the Robustness of the Relationship 52

Problems with Regression Analysis for Forecasting Costs 63

Using Regression Analysis to Forecast Sales 64

Autocorrelation in Time Series Analysis 69

Application of Regression Analysis to the Guideline Company (GC) Methods 69

Summary 73

References 74

Appendix 3A The ANOVA Table (Table A3.1, Rows 28–32) 75

Chapter 4 Annuity Discount Factors and the Gordon Model 79

Introduction 81

ADF with End-of-Year Cash Flows 83

Midyear Cash Flows 91

Starting Periods Other Than Year 1 93

Periodic Perpetuity Factors (PPFs): Perpetuities for Periodic Cash Flows 101

ADFs in Loan Mathematics 107

Relationship of the Gordon Model to the Price/Earnings and Price/Sales Ratios 110

The Bias in Annual (versus Monthly) Discounting Is Immaterial 113

Conclusions 119

References 121

Appendix 4A Mathematical Appendix 123

Appendix 4B Mathematical Appendix: Monthly ADFs 141

Part II Calculating Discount Rates 145

Chapter 5 Discount Rates as a Function of Log Size 149

Research Included in the First Edition 151

Table 5.1: Analysis of Historical Stock Returns 152

Application of the Log Size Model 167

Discussion of Models and Size Effects 181

Industry Effects 191

The Wedge between Public and Private Firm Valuations 192

Satisfying Revenue Ruling 59-60 196

Summary and Conclusions 198

References 199

Appendix 5A Automating Iteration Using Newton’s Method 203

Appendix 5B Mathematical Appendix 207

Appendix 5C Abbreviated Review and Use 211

Chapter 6 Arithmetic versus Geometric Means: Empirical Evidence and Theoretical Issues 223

Introduction 225

Theoretical Superiority of the Arithmetic Mean 226

Empirical Evidence of the Superiority of the Arithmetic Mean 227

Indro and Lee Article 232

References 233

Chapter 7 An Iterative Valuation Approach 235

Introduction 237

Equity Valuation Method 237

Invested Capital Approach 243

Log Size 245

Summary 245

References 247

Part III Adjusting for Control and Marketability 249

Chapter 8 Adjusting for Levels of Control and Marketability 253

Introduction 257

The Value of Control and Adjusting for Level of Control 257

Discount for Lack of Marketability (DLOM) 301

Conclusion 358

References 359

Appendix 8A Mathematical Appendix 365

Part IV Putting It All Together 375

Chapter 9 Empirical Testing of Abrams’s Valuation Theory 377

Introduction 379

Table 9.1: Log Size for 1938–1986 380

Table 9.2: Reconciliation to the IBA Database 382

Calculation of DLOM 387

Interpretation of the Error 400

Conclusion 401

References 401

Chapter 10 Measuring Valuation Uncertainty and Error 403

Introduction 405

Measuring Valuation Uncertainty 406

Measuring the Effects of Valuation Error 410

Summary and Conclusions 422

Reference 423

Part V Litigation 425

Chapter 11 Demonstrating Expert Bias 427

Introduction 429

Market Methods 429

A Balanced DCF Valuation 432

Summary 434

Chapter 12 Lost Inventory and Lost Profits Damage Formulas in Litigation 435

Introduction 437

Commentary to Table 12.1: Sample Damage Calculations with VM = $95 438

Table 12.1B: Lost Profits Formulas Based on EBITDA for Lost Sales on Inventory Never Produced 445

When Reality May Vary with Our Assumptions 446

Modification of Formulas for Wholesale and Retail Businesses 447

Legal Treatment 447

Summary 448

Reference 448

Part VI Valuing Esops and Buyouts of Partners and Shareholders 449

Chapter 13 ESOPs: Measuring and Apportioning Dilution 451

Introduction 453

Definitions of Dilution 454

Table 13.1: Calculation of Lifetime ESOP Costs 456

The Direct Approach 457

The Iterative Approach 466

Summary 469

References 474

Appendix 13A Mathematical Appendix 475

Chapter 14 The Trade-off in Selling to an ESOP versus an Outside Buyer 477

Section 1: Introduction 479

Section 2: Advantages and Disadvantages of Selling to an ESOP versus a Third Party 480

Section 3: The Mathematics 481

Section 4: Sample Calculations in the Tables 486

Section 5: Conclusion 494

References 494

Chapter 15 Buyouts of Partners and Shareholders 497

Introduction 499

Table 15.1: Pre- and Post-Transaction Valuations 499

Table 15.2: Dilution in FMV as a Result of the Partner Buyout 501

Sharing the Dilution 503

Conclusion 506

Part VII Probabilistic Methods 507

Chapter 16 Valuing Start-Ups 511

Issues Unique to Start-Ups 513

Organization of the Chapter 513

Part 1: First Chicago Approach 514

Venture Capital Valuation Approach 520

Part 2: Debt Restructuring Study 521

Part 3: Exponentially Declining Sales Growth Model 534

References 536

Chapter 17 Monte Carlo Risk Simulation, by Dr. Johnathan Mun 539

What Is Monte Carlo Risk Simulation? 541

Comparing Simulation with Traditional Analyses 543

Running a Monte Carlo Simulation Using Risk Simulator 543

Using Forecast Charts and Confidence Intervals 554

Tornado and Sensitivity Tools in Simulation 556

Sensitivity Analysis 563

Distributional Fitting: Single Variable and Multiple Variables 567

Getting the Risk Simulator Software 571

Chapter 18 Real Options, by Dr. Johnathan Mun 573

Part 1: Introduction to Real Options 575

Part 2: Traditional Valuation Approaches 585

Part 3: Application: Real Options SLS Software 597

Glossary 617

About the Author 621

Index 623

JAY B. ABRAMS is a leading valuation and litigation economist. He is the principal in Abrams Valuation Group, Inc. and is credited with numerous inventions, including the Abrams Log Size Model for calculating discount rates. He is also a popular finance lecturer and the author of numerous journal articles.