### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Principles-of-Cash-Flow-Valuation-pdf

### View Full Document

## 10

## 0

## Popular in Course

## Popular in Accounting

This 516 page Document was uploaded by an elite notetaker on Friday December 18, 2015. The Document belongs to a course at a university taught by a professor in Fall. Since its upload, it has received 10 views.

## Popular in Accounting

## Reviews for Principles-of-Cash-Flow-Valuation-pdf

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 12/18/15

PRINCIPLES OF CASH FLOW VALUATION AnIntegrated Market-BasedApproach This Page Intentionally Left Blank PRINCIPLES OF CASH FLOW VALUATION AnIntegrated Market -BasedApproach Joseph Tham Visiting Assistant Professor Duke Center for International Development (DCID) Sanford Institute of Public Policy Duke University Durham, North Carolina Ignacio Ve ´lez-Pareja Dean School of Industrial Engineering Politecnico Grancolombiano Bogota, Colombia AMSTERDAM •BOSTON• HEIDELBERG LONDON• NEW YORK•OXFORD PARIS SAN DIEGO SAN FRANCISCSINGAPORE•SYDNEY •TOKYO Elsevier Academic Press 200 Wheeler Road, Sixth Floor, Burlington, MA 01803, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright ▯ 2004, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333, e-mail: permissions@elsevier.com.uk. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting ‘‘Customer Support’’ and then ‘‘Obtaining Permissions.’’ Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-12-686040-8 For all information on all Academic Press publications visit our Web site at www.academicpress.com Printed in the United States of America 03 40 50 60 70 80 9 8 7 6 5 4 3 2 1 I dedicate this book to Professor Benedict Freedman, an extraordinary teacher at Occidental College. Joseph Tham I dedicate this book to my beloved Vilma, who, like an angel, gives me silent support from a distance. Ignacio Velez-Pareja This Page Intentionally Left Blank The Analyst’s Prayer Give me the intellect to seek the knowable; give me peace of mind to accept the unknowable. Give me the strength to reject the investments with negative NPV and the wisdom to select the projects with positive NPV. Most importantly, give me the inspiration (preferably the correct risk-adjusted cost of capital) to know the diﬀerence between a good investment and a bad investment. This Page Intentionally Left Blank CONTENTS PREFACE XVII ABOUT THE AUTHORS XXVII 1 Basic Concepts in Market-Based Cash Flow Valuation 1.1 Introduction 1 1.2 Market-Based Procedure for Valuation 3 1.3 Steps in Cash Flow Valuation 4 1.4 Present Value 7 1.5 The Standard After-Tax Weighted Average Cost of Capital 12 1.6 Types of Cash Flows 13 1.7 Weighted Average Cost of Capital in an M & M World 15 1.8 WACC in an M & M World with Taxes 19 1.9 The Fundamental Free Cash Flow Relationship 22 1.10 The Main Valuation Methods and Formulas for Cost of Capital 22 1.11 The Cash Flow to Equity Approach 25 1.12 Estimating the Cost of Capital26 1.13 Adjusted Present Value Approach 28 1.14 Various Formulations for the Cost of Capital 28 1.15 Summary and Concluding Remarks 28 ix x CONTENTS 2 Time Value of Money and Introduction to Cost of Capital 2.1 Introduction 31 Section 1 2.2 Nominal Prices, Constant Prices, and Real Prices38 2.3 Risk Premium with the Capital Asset Pricing Model 43 2.4 Calculating PV with a Finite Stream of Cash Flows 45 Section 2 2.5 Valuation with a Finite Stream of Cash Flows 53 2.6 Summary and Concluding Remarks 64 Appendix A A2.1 Calculating the PV with Cash Flow in Perpetuity (Without Growth) 65 A2.2 WACC in an M & M World with Taxes 70 A2.5 Free Cash Flow in Perpetuity with Growth 79 A2.6 Cash Flow to Equity 82 Appendix B B2.1 Using the CAPM to Find the Cost of Capital 84 B2.2 Discount Rate for the Tax Shield is the Return to Unlevered Equity ▯ 86 3 Basic Review of Financial Statements and Accounting Concepts 3.1 Financial Statements and Accounting Concepts 87 Section 1 3.2 Balance Sheet 89 3.3 Working Capital 91 3.4 (Book) Value of Equity 92 3.5 Income Statement 92 3.6 Cash Flow Statement (CFS) According to GAAP 96 3.7 Cash Budget Statement 98 3.8 Diﬀerences between the CFS According to GAAP and the CB Statement 103 3.9 Integration of the Financial Statements104 CONTENTS xi Section 2 3.10 Preliminary Tables 105 3.11 Summary and Concluding Remarks 110 Appendix A Appendix B 4 Constructing Integrated Pro-Forma Financial Statements, Part One 4.1 Basic Financial Statements 121 4.2 Simple Numerical Example 122 4.3 Goals and Policies for Selected Variables126 4.4 Depreciation Schedule 127 4.5 Estimated Target Variables 128 4.6 Preliminary Tables for the Simple Example 130 4.7 Constructing the Financial Statements for the Simple Example 137 4.8 Detailed Cash Budget Statement in Year 5 146 4.9 Balance Sheet 149 4.10 Cash Flow Statement According to GAAP 150 4.11 Summary and Concluding Remarks 152 Appendix A A4.1 Detailed IS for Year 1 153 A4.2 Detailed Cash Budget Statement for Year 1 154 A4.3 Detailed CB Statement for Year 2 156 5 Constructing Integrated Pro-Forma Financial Statements, Part Two 5.1 Constructing Financial Statements 159 5.2 Impact on Demand of Changes in Price and of Expenditures on Advertising and Promotion 168 5.3 Real Rate of Interest, the Risk-Premium for Debt, and the Reinvestment Return: Interest Rates Estimation 174 5.4 Depreciation Schedule 175 5.5 Initial Cash Budget for Year 0 176 5.6 Loan Schedule 177 5.7 Inventory and Quantity Purchased 178 5.8 Relationship between the Quantity Purchased and the Purchase Price 179 xii CONTENTS 5.9 Cost of Goods Sold 181 5.10 Selling and Administrative Expenses 182 5.11 Receivables and Payables 182 5.12 The Logic of the Model 183 5.13 Constructing Financial Statements for the Complex Example 185 5.14 Summary and Concluding Remarks 192 Appendix A A5.1 A Brief Introduction to Price-Demand Elasticity193 Appendix B B5.1 A Detailed Spreadsheet Example 199 6 The Derivation of Cash Flows Section 1 6.1 Derivation of Free Cash Flows 229 6.2 The Fundamental Free Cash Flow Relationship 230 Section 2 6.3 Deriving the FCF from the CB Statement 235 6.4 Total CCF Versus Operating CCF 242 6.5 Deriving the FCF from the CFS According to GAAP 248 6.6 Deriving the FCF from the EBIT in the IS 249 6.7 Deriving the CFE from the NI 253 6.8 Advantages of Using the Cash Budget Approach 254 6.9 Summary and Concluding Remarks 255 Appendix A A6.1 Calculating Excess Cash from the IS and the BS 256 Appendix B B6.1 Deriving the FCF from NI 258 B6.2 Deriving the CCF from NI 259 7 Using the WACC in Theory and Practice Section 1 7.1 Approaches to the Cost of Capital 261 CONTENTS xiii Section 2 7.2 Review of Basic Ideas 263 Section 3 7.3 Three Simple Expressions for the Cost of Capital269 Section 4 7.4 General Framework 273 Section 5 7.5 Numerical Examples with the Complex Example 278 7.6 Summary and Concluding Remarks 283 Appendix A A7.1 Algebraic Approach 284 Appendix B B7.1 Using the CAPM to Find the Cost of Capital 290 Appendix C C7.1 The Calculation of Levered Values with OCFs 292 8 Estimating the WACC for Non-Traded Firms 8.1 Practical Approaches to the Cost of Capital for Traded and Non-Traded Firms 297 8.2 Finding the Relationship Between Levered and Unlevered Betas 298 8.3 Valuation for Traded Firms 301 8.4 Valuation for Non-Traded Firms 305 8.5 Summary and Concluding Remarks 324 Appendix A A8.1 Estimation with Total Risk 325 Appendix B Appendix C 9 Beyond the Planning Period: Calculating the Terminal Value 9.1 Introduction 333 9.2 Operation Versus Liquidation 334 xiv CONTENTS 9.3 Calculating the Salvage (or Liquidation) Valu335 9.4 Terminal Value Versus Salvage Value 335 9.5 The Standard Approach for Estimating the TV 336 9.6 TV Calculated for the Discounted Cash Flow Method Using NOPLAT and Constant Leverage 342 9.7 Amount of Reinvestment for Growth in the FCF 343 9.8 Calculating the ROMVIC 344 9.9 Additional Adjustments 345 9.10 A Comment on Constant Leverage 346 9.11 Calculating TV with the CCF and APV Approaches 350 9.12 Summary and Concluding Remarks 350 Appendix A A9.1 TV and Amount of Revinvestment 351 A9.2 Return on the Market Value of Invested Capital (MVIC351 A9.3 Additional Reinvestment for Higher Growth of NOPLAT 352 Appendix B B9.1 Independent Calculation of the TV Levered Value 354 B9.2 Value of Debt in the Terminal Period N 354 B9.3 Value of the Tax Shield in the Terminal Year N355 B9.4 Levered Value in the Terminal Year N 355 B9.5 TV with CCF and APV 358 Appendix C C9.1 Pure FCF and Non-Pure FCF 360 Appendix D D9.1 Matching the Residual Income Method, EVA▯ and DCF 363 D9.2 What Is Value Added? 363 D9.3 Assumptions and Financial Statements for the Complex Example 365 D9.4 Summary and Concluding Remarks 370 10 Theory for Cost of Capital Revisited 10.1 Cost of Capital with a Finite Stream of Cash Flows371 10.2 Numerical Example 375 10.3 Loan Schedule with Constant Leverage 382 10.4 Policy on Debt Financing and the WACC in the Presence of Taxes 384 CONTENTS xv 10.5 The Miles and Ezzel (M & E) WACC 386 10.6 WACC for a Finite Stream of Cash Flows in an M & M World with Taxes 388 10.7 Fixed Percentage of Debt 390 10.8 Theory on the Cost of Capital Applied to Finite Cash Flows 393 10.9 Standard WACC Applied to the CCF 395 10.10 Standard After-Tax WACC Applied to the FCF 396 10.11 Return to (Levered) Equity 398 10.12 Alternative Adjusted WACC Applied to the FCF (Optional) 400 10.13 Alternative WACC Applied to the CCF (Optional) 402 10.14 Numerical Example 403 10.15 Adjusted Present Value (APV) Approach 405 10.16 Summary and Concluding Remarks 408 Appendix A A10.1 Derivation of the M & E WACC 410 Appendix B B10.1 Numerical Example with Constant Leverage 413 B10.2 Basic Information 415 B10.3 WACC Applied to the CCF 417 B10.4 Standard After-Tax WACC Applied to the FCF 417 B10.5 Alternative Adjusted WACC for the FCF 418 11 How Are Cash Flows Valued in the Real World 11.1 Introduction 419 11.2 Case study: TIMANCO S. A. E. S. P. Empresa de Telecomunicaciones 427 11.3 Valuation Methods Based on Book Values 449 11.4 Concluding Remarks 458 Appendix A A11.1 The Practitioner’s Point of View459 Appendix B B11.1 Estimated Macroeconomic Variables 462 B11.2 Construction Period 462 B11.3 Installation Program 463 B11.4 Operation of the Project 463 B11.5 Consumption Estimation 466 xvi CONTENTS B11.6 Payroll Expenses 466 B11.7 Operating Costs and Expenses 467 B11.8 Financial Statements 468 SELECTEDBIBLIOGRAPHYANDREFERENCES 479 INDEX 483 PREFACE Mathematics (Valuation?) may be deﬁned as the subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell (1872 to 1970) Our main objective is to present clearly and rigorously the general principles for ﬁnite cash ﬂow valuation, with a balanced mix of theoretical ideas and practical illustrations. Roughly speaking, to value a ﬁnite stream of cash ﬂows, we must complete two activities. First, we must estimate the expected annual cash ﬂows, and second, we must determine the appropriate risk- adjusted discount rate(s) for the cash ﬂow. Principles of Cash Flow Valuation: An Integrated Market-Based Approach provides a comprehensive and practical market-based framework for the valuation of ﬁnite cash ﬂows that are derived from a set of integrated ﬁnancial statements, namely, the income statement, the balance sheet, and the cash budget (which has been specially adapted for valuation purposes). In addition to the standard cash ﬂows, such as the free cash ﬂow (FCF) and the cash ﬂow to equity (CFE), the book discusses the novel capital cash ﬂow (CCF) approach to valuation. With the CCF approach, the tax shield from debt ﬁnancing is added directly to the FCF, and there is no need to adjust the weighted average cost of capital (WACC). Detailed numerical examples illustrate the construction of the pro- forma ﬁnancial statements, step by step. A separate chapter is devoted to the calculations on the terminal value. To meet the needs of readers with diﬀerent backgrounds in ﬁnance, we present the theory on the cost of capital in a progressive spiral fashion, from the simple and intuitive to the complex, with numerous numerical examples to facilitate understanding and learning. Assuming perfect capitals, we derive the appropriate expressions for the WACC, as applied to the FCF and CCF, and carefully explain the subtleties that underlie the diﬀerent formulations of the WACC in relationship to the xvii xviii PREFACE risk of the tax shield. In particular, we show how to solve the ‘‘circularity’’ problem in the estimation of the WACC with market values. In addition, with the appropriate assumptions, we show that the results from the dis- count cash ﬂow (DCF) methods are fully consistent with the Residual Income Model (RIM) and the Economic Value Added (EVA ) approach. 1 We discuss the valuation of both traded and non-traded ﬁrms, and using a real-life detailed case study, we illustrate and calculate the discrepancies that arise in the valuation of cash ﬂows, depending on the assumptions that are made in the formulation of the WACC. To appeal to a wide audience, the level of the mathematics has been kept to a minimum. It would be helpful if the reader has taken some accounting courses, but prior knowledge of accounting principles is not necessary for reading the book. We use and explain the basic accounting knowledge that are needed for the purposes of the book. The book will be of interest to a wide range of readers, from students of applied ﬁnance to practitioners and experts who would like to gain a deeper appreciation for the theory on the cost of capital in the context of ﬁnite cash ﬂow valuation. OVERALLFEATURE: The book presents an integrated approach to ﬁnite cash ﬂow valuation that combines the ﬁnancial statements with the theory on the cost of capital. The following details some key features of the book. Simplifying reality It is hardly a profound truism to state that reality is complex. We recognize that reality is complex, and consequently, we have to model the key features of reality by making relevant simplifying assumptions. We progress from the simple to the complex. The level of complexity depends on the availability of information and the purpose of the analysis. Active reading and learning The readers are encouraged to read and learn actively by reconstructing the ﬁnancial statements for themselves on a spreadsheet. Wherever possible, we have provided suﬃcient detail for the readers to do so. If there are obscu- rities or ambiguities in the explanations given in the text and readers 1The reconciliation of the DCF methods with the RIM and EVA approaches is a good check on the results of the valuation exercise. PREFACE xix encounter diﬃculties in replicating the tables in the text, they are encouraged to e-mail the authors for assistance and clariﬁcation. We shall respond as promptly as we can. (See p. xxiii for contact information.) Finite stream of cash £ows and perpetuities Intentionally and for reasons of practicality, whenever possible, we avoid the use of cash ﬂows in perpetuities and present examples with ﬁnite periods. Admittedly, the use of cash ﬂows in perpetuities greatly simpliﬁes the expo- sition on the cost of capital; however, the loss in ﬂexibility is too high a price to pay for the simplicity. In general, for the forecast or planning period, we use ﬁnite streams of cash ﬂows. In the estimation of the (present) value in the terminal period, based on the cash ﬂows that occur beyond the planning period, we use perpetuities. Duration of the analysis and calculation of the terminal value With suﬃcient information, we can use spreadsheets to construct ﬁnancial statements of reasonable lengths such as ten to ﬁfteen years. The longer the period of analysis, the greater is the uncertainty in the expected values of the key parameters in the model. Typically, it may be more common for the forecast period to be ﬁve to eight years. For greatest ﬂexibility, we specify a ﬁnite stream of cash ﬂows for the forecast period and use perpetuities only for the calculation of the terminal value at the end of the forecast period. If the forecast period is short (e.g., ﬁve years), then the contribution of the terminal value to the overall value may be substantial and the proper calculation of the terminal value may be of high importance. The following section is a brief description of the organization and content of the book. In valuation, one of the key concepts is the cost of capital. Since it is neither possible nor advisable to present the theory on the cost of capital in one chapter, we discuss the cost of capital over several chapters with increasing degrees of complexity. In Chapters One and Two, we introduce the basic ideas related to cost of capital. In Chapter Seven we apply the ideas on cost of capital to the speciﬁc numerical example from Chapter Five, and in Chapter Ten, we revisit the theory on the cost of capital at a more advanced level. ORGANIZATIONOF THE BOOK . Chapter One provides a clear and succinct qualitative overview on the theory of the cost of capital in the context of ﬁnite cash ﬂow valuation. xx PREFACE . Chapter Two is a review of discounting and presents some numerical examples on cost of capital. . Chapter Three is a review of some basic ideas in accounting and ﬁnancial statements. . Chapters Four and Five present detailed numerical illustrations on the construction of an integrated set of ﬁnancial statements for cash ﬂow valuation without terminal value. The estimation of the terminal value is presented in Chapter Nine. . Chapter Six shows the derivations of various discounted cash ﬂow methods: FCF, CFE, and CCF. . Chapters Seven and Ten provide detailed presentations on diﬀerent formulations for the WACC applied to the FCF and CCF. . Chapter Eight discusses the cost of capital for non-traded ﬁrms and presents methods for valuing them. . Chapter Nine discusses the estimation of the terminal value. The appendix to Chapter Nine demonstrates the equivalence between the ▯ DCF methods, and RIM and EVA . . Chapter Eleven shows how to apply the integrated approach and the theory on the cost of capital to a real-life case study. CONTENTOF THE BOOK Chapter One Chapter One begins with a brief discussion on market-based valuation, the motivation for investment, and the time value of money. We discuss the relevance of perfect capital markets and the absence of arbitrage for valua- tion, and we examine the estimation of the cost of capital and valuation of 2 cash ﬂows in perfect capital markets. We deﬁne the diﬀerent types of cash ﬂows and the cost of capital in the Modigliani and Miller (M & M) world, with and without taxes, where roughly speaking, the M & M world is a world with perfect capital markets. 3 In particular, we present the standard FCF, the cash ﬂow to debt (CFD), CFE, and the alternative CCF. There is a technical deﬁnition for ‘‘free cash ﬂow,’’ and later in Chapter Six, we deﬁne and discuss the annual FCF more precisely. After discussing the theory of valuation in M & M 2Clearly, the analysis with perfect capital markets is unrealistic. It is a point of departure for beginning the valuation exercise and provides a tractable framework for thinking about the relevant issues. 3The M & M world is named in honor of Modigliani and Miller, who made the funda- mental and original contributions to cost of capital and valuation. 4 For the curious reader, the free cash ﬂow (FCF) is the sum of the cash ﬂow to equity (CFE) and the cash ﬂow to debt (CFD), less the tax shield (TS). In symbols, FCF ¼CFEþCFD▯TS. PREFACE xxi worlds with and without taxes, we present diﬀerent valuation methods with the corresponding formulations for the cost of capital. We believe that an early introduction to the cost of capital provides the proper motivation and framework for studying the concepts in the later chapters. Chapter Two Chapter Two reviews the time value of money, inﬂation, and ideas related to discounting. Also, we provide an introduction to the cost of capital with ﬁnite cash ﬂows. Formulas for cost of capital with cash ﬂows in perpetuity are easier to understand than ﬁnite cash ﬂows, and typically, the formulas that are derived from cash ﬂows in perpetuity are applied to ﬁnite cash ﬂows. Since the emphasis in this book is on ﬁnite cash ﬂows, for consistent results, we present numerical examples with ﬁnite cash ﬂows. Moreover, it is very likely that the reader is familiar with and has been exposed to the cost of capital as applied to cash ﬂows in perpetuity. In the appendix to Chapter Two, for completeness and reference, we present detailed6numerical examples with cash ﬂows in perpetuity with and without growth. This Appendix is for those readers who wish to read an elementary exposition on cost of capital with cash ﬂows in perpetuity, which may facilitate the transition to cost of capital applied to ﬁnite cash ﬂows. Chapter Three Chapter Three presents a basic review of ﬁnancial statements and account- ing concepts. To be speciﬁc, it discusses the balance sheet, the income statement, the cash ﬂow statement according to Generally Accepted Accounting Principles (GAAP), and the annual cash budget statement. For most readers, this chapter should be a review and serve as a reference chapter for the ideas that are discussed in subsequent chapters. Chapters Four and Five Chapters Four and Five present detailed numerical examples to illustrate the construction of ﬁnancial statements, namely the income statement (IS), the 5The reader can conﬁrm that when the formulas for cash ﬂows in perpetuity are applied to ﬁnit6 cash ﬂows, the results are inconsistent. We stress that the results and formulas with cash ﬂows in perpetuity may be misleading if they are applied to ﬁnite cash ﬂows. We remind the reader that we derive ﬁnite cash ﬂows from ﬁnancial statements; we do not derive cash ﬂows in perpetuity from ﬁnancial statements. xxii PREFACE balance sheet (BS), and the cash ﬂow statement (CFS) for the planning or forecast period, which we assume is ﬁve years. We present the cash ﬂow statement according to GAAP. It is important to note that we construct the ﬁnancial statements in nominal terms and the costs of capital are also in nominal terms. Both Chapters Four and Five are self-contained. To provide the reader with a strong understanding of the basic determinants of the ﬁnancial statements and the cash ﬂows that are derived from the ﬁnancial statements, in our models, we use disaggregated variables such as the prices and quan- tities of the inputs and outputs rather than aggregated line items such as earnings before interest and taxes (EBIT) or net income (NI). Once the reader understands these models that are based on the disaggregated para- meters, she may wish to simplify and model more aggregated line items. With the disaggregated models, it is feasible and convenient to conduct sensitivity analysis on the desired outcomes of the model with respect to changes in the key parameters. To meet the needs of readers with diverse backgrounds, we present two chapters with diﬀerent levels of diﬃculty. Chapter Five extends the simple example in Chapter Four by introducing additional assumptions and com- plexities into the model. The reader with a strong background should proceed directly to Chapter Five. Other readers may wish to begin with Chapter Four and then proceed to Chapter Five. Chapter Four proceeds at a slower pace and provides explanations in greater detail to ensure that all readers can follow the material. We link the construction of the IS and the BS with the cash budget (CB) statement. Typically the CB is a monthly statement that is an important tool for cash management. Here, we employ an annual CB statement and inte- grate it with the IS and the BS. For ease in exposition, the CB statement is referred to as a ﬁnancial statement, even though according to GAAP the CB statement is not a ﬁnancial statement. We emphasize that we present a valuation framework that integrates all of the four statements: the income statement, the balance, the cash ﬂow statement and the cash budget statement. Chapter Six In the ﬁrst part of Chapter Six, using a new and simple approach, we derive the FCF statement from the CB statement. For the moment, the terminal value is not included in the calculation of the FCF. The second part of Chapter Six describes the typical derivation of the FCF from the income statement with the indirect method. Rather than specifying the FCF directly without reference to the ﬁnan- cial statements, we propose to derive the FCF statement from the CB PREFACE xxiii statement, which in turn is linked to the IS and BS. This is a distinctive feature of the integrated approach that we present. We prefer this approach because the CB is closest to the idea of free cash ﬂow. In fact, the CB records all the cash movements of a ﬁrm or a project. With this integrated approach, all four ﬁnancial statements—IS, BS, CB, and FCF statements—are linked, and we can easily check for internal consistency. 7 With this approach, we ‘‘see’’ most of the items that are considered as part of the FCF and the probability of mistakes in the construction of the FCF is reduced. The only items that are not included explicitly in the CB are the tax adjustment or tax savings, the opportunity costs and the terminal value in the case of a going concern. We postpone the discussion on the calculation of the terminal value to Chapter Nine. By using the annual CB to derive the FCF statement, we do not disregard the CB, which is a very useful managerial tool for cash management. We expect the reader will ﬁnd this approach more intuitive and easier to follow than the typical derivation with the indirect method from the earnings before EBIT or NI in the income statement. The reader may decide for herself after she has compared all the diﬀerent approaches. With the availability of appropriate software, the cash bud- get can be calculated on a weekly or monthly basis. Chapter Seven Chapter Seven focuses on the practical issues that are related to the estima- tion of the cost of capital and applies the ideas on the cost of the capital to the cash ﬂows that have been derived from the ﬁnancial statements that were constructed for the complex example in Chapter Five. There is an inherent circularity in the valuation process. To estimate the WACC, we require the market values of debt and equity; to estimate the market values of debt and equity, we must know the WACC. We show how to use the spreadsheet to solve the circularity problem. 8 Chapter Eight Chapter Eight is about various methods for estimating the cost of capital for non-traded ﬁrms. 7Moreover, by using all the ﬁnancial statements, the integrated approach bridges and (hopefully) narrows the gap between the ‘‘accounting’’ frame of mind and the ‘‘valuation’’ frame of m8nd. In using the CCF approach and assuming that the value of discount rate for the tax shield is equal to the return to unlevered equity, there is no circularity. xxiv PREFACE Chapter Nine Chapter Nine examines the estimation of the terminal value and value the FCF for the complex example in Chapter Five. In addition, we show that the results with DCF methods, RIM, and EVA ▯ are fully consistent. Chapter Ten Using ﬁnite streams of cash ﬂows, Chapter Ten revisits the theory on the cost of capital and examines some of the underlying assumptions in more detail and depth. Chapter Eleven Chapter Eleven applies the ideas on the cost of capital to a real case study and show the discrepancies that arise from the diﬀerent assumptions with respect to the estimation of the cost of capital. We hope the ideas in the book are useful in your studies, work, and professional life and you enjoy reading the book as much as we have enjoyed writing it. Our papers on valuation are available on the Social Science Research Network (SSRN) at papers.ssrn.com. Also, see Ignacio’s website at http://www.poligran.edu.co/decisiones. We must mention that the writing of the book across continents (Joe in Ho Chi Minh City, New York City, and Boston, and Ignacio in Bogota) ´ would have been impossible without the Internet, which permitted the innumerable e-mails and the long, heated discussions with instant messa- ging. We welcome critical feedback and constructive comments and have tried our best to eliminate errors and typos. However, we are almost certain that there are errors and mistakes that are just waiting to be discovered and would appreciate notiﬁcation of them. ACKNOWLEDGMENTS AND SPECIALTHANKS We thank Nicholas Wonder, Andreas Loeﬄer, Paul Fieten, Shikhar Ranjan, Gordon Sick, and Pablo Fernandez for very helpful correspondence and assistance on topics related to cash ﬂow valuation and cost of capital. Also, Joe wishes to thank all the participants and members of the Teaching Team in Project Appraisal at the Fulbright Economics Teaching Program (FETP) in Ho Chi Minh City, Vietnam, who had to suﬀer through the initial drafts of my unintelligible notes on cost of capital. In particular, I would like to thank Lora Sabin, Cao Hao Thi, Le Thi Thanh Loan and Ngo PREFACE xxv Kim Phuong for their comments and feedback, which forced me to clarify my own thinking and improve the notes. Le Thi Thanh Loan’s careful reading helped us to eliminate many errors. Joe also wishes to thank the many groups of participants, colleagues, tutors, and lecturers in the Program on Investment Appraisal and Manage- ment (PIAM) at the Harvard Institute for International Development (HIID). For their support in various ways, Joe thanks G.P. Shukla, Graham Glenday, Mano Ranjan, Alberto Barreix, Baher El-Hifnawi, Rags Narain, Migara Jayawardena, Savvakis Savvides, Ranjay Gulati, Farid Bassiri, Hien, Little Trang, Jim Kass, Asad Jumabhoy, and Feryal Jumabhoy. Joe thanks Jonathan Haughton for having conﬁdence in him and giving him his ﬁrst break. Special thanks must go to Baher (a.k.a. the chief) for unknowingly (and inadvertently) provoking me along an intellectual path that has led to the writing of the book. Ignacio wishes to thank Julio Sarmiento at Universidad Javeriana, Bogota, Colombia, for his interest in reading and commenting on drafts and using them in teaching. Special thanks to Ramiro de la Vega and Guillermo Rossi, private ﬁnancial consultants for reading drafts and colla- borating in the writing of Chapter Eleven. Also, Ignacio thanks all his undergraduate and graduate students at Politecnico Grancolombiano and Universidad Javeriana at Bogota, C o l o m - bia, and ICESI, Cali, Colombia, who suﬀered the initial versions of many chapters, all written in a foreign language. At Academic Press, Scott Bentley, Karen Maloney, and Jane Mac- Donald have been most supportive and encouraging. In addition, Dennis McGonagle, Mamata Reddy, Julio Esperas, and Jaya Nilamani and her team at Integra-India have taken every eﬀort to produce the best book. Special and deep-felt thanks to Jennifer Mulik, Kate Laurence, Nancy Scott, and David Trzcinski for assisting in the construction of the index during the crunch time. Also, Joe wishes to thank everyone at the Center for International Health and Development (CIHD) at the Boston University School of Public Health (BUSPH) for their understanding, tolerance and moral support. Finally, we would like to give our deepest thanks to the anonymous reviewers who were generous in their constructive comments, which greatly improved the chapters in the book. We have tried to follow most of their recommendations and accept responsibility for all remaining errors. The ﬁnal test of the relevance of the book is the approval of the reader. Again, we look forward to critical suggestions for improvement and construc- tive feedback from readers. Please contact Joe at ThamJx@yahoo.com or Ignacio at ivelez98@yahoo.com to provide feedback. This Page Intentionally Left Blank ABOUT THE AUTHORS Joseph Tham is visiting assistant professor at the Duke Center for International Develop- ment (DCID), Terry Sanford Institute of Public Policy at Duke University and a research associate at the Center for International Health and Development (CIHD) at Boston University School of Public Health (BUSPH). He is an applied economist with interests in the ﬁnancial, economic, and risk analysis of social sector projects. From 1996 to 2001, he was a lecturer at the Fulbright Economics Teaching Program (FETP) in Ho Chi Minh City, Vietnam, an education program managed by Harvard University. He also served as a project associate at the Center for Business and Government (CBG), John F. Kennedy School of Govern- ment at Harvard University (2000–2001). For several years, he was a lecturer in the Program on Investment Appraisal and Management (PIAM) at the Harvard Institute for International Development (HIID) and spent a year in Indonesia, analyzing educational ﬁnance for the Asian Development Bank. He holds a bachelor’s degree in mathematics from Occidental College and a doctorate in education from Harvard University. xxvii xxviii ABOUT THE AUTHORS Ignacio Velez-Pareja is Dean of the School of Industrial Engineering at Politecnico Grancolombiano in Bogota, Colombia. His interests have been in the areas of Investment Decision Analysis and Operations Research. Recently, he has worked mainly in the ﬁnan- cial arena. Half of his professional life has been spent in private industry and the other half in academia at the most important universities in Colombia. He has served in diﬀerent functional areas for private ﬁrms, including top management positions and as head of Industrial Engineering at Universidad de los Andes and Business Administration at Universidad Javeriana in Bogota, C o l o m b i a . He holds a master’s degree in industrial engineering from the University of Missouri at Columbia and has published in international and local journals. He is the author of two books, Decisiones de inversion: enfocado a la valoracion ´ de empresas and Decisiones empresariales bajo riesgo e incertidumbre, a n d h a s published more than 50 articles on management, ﬁnance, scientiﬁc research, education, and labor issues and music in local and international peer-reviewed academic publications such as Cuadernos de Administracion, Interfaces, ´ Latin American Business Review, Monografıas Facultad de Administracion ´ (Universidad de los Andes) and Academia, Revista Latinoamericana de Administracion, CLADEA. His academic writings can be found at http://www.poligran.edu.co/decisiones. ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯ ▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.