Corporate Finance, Eleventh Edition

Corporate Finance, Eleventh Edition wWó Z^ l {2 3`, €0rpDorate Tinance Corporate Finance ros61752 fm i xxxv indd 1 10/08/15 7 03 PM The McGraw Hill/Irwin Series in Finance, Insurance, and Real Estate[.]

Corporate Finance

Stephen A Ross

Franco Modigliani Professor of Finance and Economics Sloan School of Management

Massachusetts Institute of Technology

Consulting Editor

FINANCIAL MANAGEMENT

Block, Hirt, and Danielsen

Foundations of Financial Management

Fifteenth Edition

Brealey, Myers, and Allen

Principles of Corporate Finance

Eleventh Edition

Brealey, Myers, and Allen

Principles of Corporate Finance, Concise

Second Edition

Brealey, Myers, and Marcus

Fundamentals of Corporate Finance

Cornett, Adair, and Nofsinger

Finance: Applications and Theory

Grinblatt and Titman

Financial Markets and Corporate Strategy

Ross, Westerfield, Jaffe, and Jordan

Corporate Finance: Core Principles

First Edition

White Financial Analysis with an Electronic Calculator

Seventh Edition

Stewart, Piros, and Heisler Running Money: Professional Portfolio Management

Rose and Hudgins Bank Management and Financial Services

Allen, Melone, Rosenbloom, and Mahoney Retirement Plans: 401(k)s, IRAs, and Other Deferred Compensation Approaches

Eleventh Edition

Altfest Personal Financial Planning

Brief Contents

Part I

OVERVIEW

1 Introduction to Corporate Finance 1

2 Financial Statements and Cash Flow 20

3 Financial Statements Analysis and Financial Models 44

Part II

VALUATION AND CAPITAL BUDGETING

4 Discounted Cash Flow Valuation 87

5 Net Present Value and Other Investment Rules 135

6 Making Capital Investment Decisions 171

7 Risk Analysis, Real Options, and Capital Budgeting 208

8 Interest Rates and Bond Valuation 238

Part III

RISK

10 Risk and Return: Lessons from Market History 302

11 Return and Risk: The Capital Asset Pricing Model (CAPM) 331

12 An Alternative View of Risk and Return: The Arbitrage Pricing Theory 374

13 Risk, Cost of Capital, and Valuation 396

Part IV

CAPITAL STRUCTURE AND DIVIDEND POLICY

14 Efficient Capital Markets and Behavioral Challenges 431

15 Long-Term Financing: An Introduction 471

16 Capital Structure: Basic Concepts 490

17 Capital Structure: Limits to the Use of Debt 522

18 Valuation and Capital Budgeting for the Levered Firm 555

19 Dividends and Other Payouts 577

Part V

LONG-TERM FINANCING

OPTIONS, FUTURES, AND CORPORATE FINANCE

22 Options and Corporate Finance 677

23 Options and Corporate Finance: Extensions and Applications 722

24 Warrants and Convertibles 746

25 Derivatives and Hedging Risk 767

31 International Corporate Finance 939

Appendix A: Mathematical Tables 966 Appendix B: Solutions to Selected End-of-Chapter Problems 975 Appendix C: Using the HP 10B and TI BA II Plus

Subject Index 1001

PART I Overview

Chapter 1

1.1 What Is Corporate Finance? 1

1.2 The Corporate Firm 4

1.3 The Importance of Cash Flows 8

1.4 The Goal of Financial Management 11

1.5 The Agency Problem and Control

2.1 The Balance Sheet 20

2.2 The Income Statement 23

Generally Accepted Accounting Principles 24

2.4 Net Working Capital 28

2.5 Cash Flow of the Firm 29

2.6 The Accounting Statement of Cash Flows 32

2.7 Cash Flow Management 34

Mini Case: Cash Flows at Warf Computers, Inc 42

Chapter 3Financial Statements Analysis

3.1 Financial Statements Analysis 44

3.2 Ratio Analysis 48

Short-Term Solvency or Liquidity Measures 49

Asset Management or Turnover Measures 52

3.3 The DuPont Identity 58

Problems with Financial Statement Analysis 60

3.4 Financial Models 61

3.5 External Financing and Growth 67

A Note about Sustainable

3.6 Some Caveats Regarding Financial

Mini Case: Ratios and Financial Planning at

PART II Valuation and Capital Budgeting

Chapter 4

4.1 Valuation: The One-Period Case 87 4.2 The Multiperiod Case 91

4.3 Compounding Periods 102

Distinction between Annual Percentage Rate

4.6 What Is a Firm Worth? 120

Appendix 4A: Net Present Value:

First Principles of Finance 134

Appendix 4B: Using Financial Calculators 134

Chapter 5

Net Present Value and Other

5.1 Why Use Net Present Value? 135

5.2 The Payback Period Method 138

5.3 The Discounted Payback Period Method 141

5.4 The Internal Rate of Return 141

5.5 Problems with the IRR Approach 145

Definition of Independent and Mutually

Two General Problems Affecting Both Independent and Mutually Exclusive Projects 145 Problems Specific to Mutually Exclusive Projects 149

5.6 The Profitability Index 155

5.7 The Practice of Capital Budgeting 157

Chapter 6Making Capital Investment Decisions 171

6.1 Incremental Cash Flows: The Key

6.2 The Baldwin Company: An Example 174

6.3 Alternative Definitions

of Operating Cash Flow 181

6.4 Some Special Cases of Discounted Cash Flow Analysis 184

Investments of Unequal Lives: The Equivalent

6.5 Inflation and Capital Budgeting 190

Chapter 7Risk Analysis, Real Options,

7.1 Sensitivity Analysis, Scenario Analysis, and Break-Even Analysis 208

Sensitivity Analysis and Scenario Analysis 208

7.2 Monte Carlo Simulation 216

Step 2: Specify a Distribution

Step 3: The Computer Draws

7.4 Decision Trees 225

Chapter 8

8.1 Bonds and Bond Valuation 238

Finding the Yield to Maturity:

8.4 Inflation and Interest Rates 257

Inflation Risk and Inflation-Linked Bonds 258

8.5 Determinants of Bond Yields 261

The Term Structure of Interest Rates 261

Bond Yields and the Yield Curve:

Mini Case: Financing East Coast Yachts’s

9.1 The Present Value of Common Stocks 273

Valuation of Different Types of Stocks 274

9.2 Estimates of Parameters in the Dividend Discount Model 278

Dividends or Earnings: Which to Discount? 282

9.4 Valuing Stocks Using Free Cash Flows 287 9.5 The Stock Markets 288

Mini Case: Stock Valuation at Ragan Engines 300

PART III Risk

Chapter 10 Risk and Return: Lessons

and Risk-Free Returns 314 10.5 Risk Statistics 314

Normal Distribution and Its Implications

10.6 More on Average Returns 318

Arithmetic versus Geometric Averages 318 Calculating Geometric Average Returns 318 Arithmetic Average Return or Geometric

10.7 The U.S Equity Risk Premium: Historical and International Perspectives 320

Mini Case: A Job at East Coast Yachts 329

Chapter 11

Return and Risk: The Capital Asset

11.1 Individual Securities 331

11.2 Expected Return, Variance,

11.3 The Return and Risk for Portfolios 337

Variance and Standard Deviation of a Portfolio 338

11.4 The Efficient Set for Two Assets 341

11.5 The Efficient Set for Many Securities 346

Variance and Standard Deviation in a

11.6 Diversification 349

The Anticipated and Unanticipated

11.7 Riskless Borrowing and Lending 352

11.8 Market Equilibrium 355

Definition of the Market Equilibrium Portfolio 355 Definition of Risk When Investors

11.9 Relationship between Risk and Expected

Expected Return on Individual Security 360

Mini Case: A Job At East Coast

Appendix 11A: Is Beta Dead? 373

Chapter 12

An Alternative View of Risk and Return:

12.1 Introduction 374

12.2 Systematic Risk and Betas 374

12.4 Betas, Arbitrage, and Expected Returns 382

The Market Portfolio and the Single Factor 383

12.5 The Capital Asset Pricing Model and the Arbitrage Pricing Theory 384

Mini Case: The Fama–French Multifactor

Chapter 13Risk, Cost of Capital,

13.1 The Cost of Capital 396 13.2 Estimating the Cost of Equity

Capital with the CAPM 397

13.5 The Dividend Discount

13.6 Cost of Capital for Divisions

13.7 Cost of Fixed Income Securities 411

13.8 The Weighted Average Cost of Capital 413 13.9 Valuation with R WACC 414

13.10 Estimating Eastman Chemical’s Cost of Capital 418 13.11 Flotation Costs and the Weighted

Average Cost of Capital 420

Summary and Conclusions 423

Mini Case: Cost of Capital for Swan Motors 429

Appendix 13A: Economic Value Added and

the Measurement of Financial

PART IV Capital Structure

and Dividend Policy

Chapter 14

Efficient Capital Markets

14.1 Can Financing Decisions Create Value? 431

14.2 A Description of Efficient Capital Markets 433

14.3 The Different Types of Efficiency 436

Some Common Misconceptions about the

14.4 The Evidence 440

14.5 The Behavioral Challenge

to Market Efficiency 445

Independent Deviations from Rationality 447

14.6 Empirical Challenges to Market Efficiency 449

14.7 Reviewing the Differences 454

14.8 Implications for Corporate Finance 456

1 Accounting Choices, Financial Choices,

3 Speculation and Efficient Markets 458

Mini Case: Your 401(k) Account at

Chapter 15

Long-Term Financing: An Introduction 471

15.1 Some Features of Common

and Preferred Stocks 471

Which Are Best: Book or Market Values? 486

Chapter 16

16.1 The Capital Structure Question and the Pie Theory 490 16.2 Maximizing Firm Value versus

Maximizing Stockholder Interests 491 16.3 Financial Leverage and Firm Value:

Leverage and Returns to Shareholders 493

Expected Return and Leverage

The Weighted Average Cost

of Capital, RWACC, and Corporate Taxes 512 Stock Price and Leverage under

Mini Case: Stephenson Real Estate Recapitalization 521

Chapter 17Capital Structure: Limits to

17.1 Costs of Financial Distress 522

17.4 Integration of Tax Effects

and Financial Distress Costs 532

17.6 Shirking, Perquisites, and Bad

Investments: A Note on Agency Cost of Equity 536

Effect of Agency Costs of Equity

17.7 The Pecking-Order Theory 539

17.8 Personal Taxes 542

The Effect of Personal Taxes on Capital Structure 542

17.9 How Firms Establish Capital Structure 544

Mini Case: Mckenzie Corporation’s

Appendix 17A: Some Useful Formulas of

Appendix 17B: The Miller Model and the

Chapter 18

Valuation and Capital Budgeting

18.1 Adjusted Present Value Approach 555

18.2 Flow to Equity Approach 557

Step 1: Calculating Levered

and WACC Approaches 559

18.5 Valuation When the Discount Rate

Must Be Estimated 562

Mini Case: The Leveraged Buyout

Appendix 18A: The Adjusted Present

Value Approach to Valuing Leveraged Buyouts 576

Chapter 19

19.1 Different Types of Payouts 577 19.2 Standard Method of Cash

Dividend Payment 578 19.3 The Benchmark Case: An Illustration

of the Irrelevance of Dividend Policy 580

Current Policy: Dividends Set Equal

Alternative Policy: Initial Dividend Is Greater

Firms with Sufficient Cash to Pay a Dividend 588

19.6 Real-World Factors Favoring

a High-Dividend Policy 591

Information Content of Dividends

19.7 The Clientele Effect: A Resolution

of Real-World Factors? 596 19.8 What We Know and Do Not Know about Dividend Policy 598

Some Survey Evidence about Dividends 601

19.9 Putting It All Together 603 19.10 Stock Dividends and Stock Splits 605

and Stock Dividends 605

Value of Stock Splits and Stock Dividends 607

Venture Capital Investments

20.2 The Public Issue 621

20.3 Alternative Issue Methods 622

20.4 The Cash Offer 624

Underpricing: A Possible Explanation 629

20.5 The Announcement of New Equity

and the Value of the Firm 631

20.6 The Cost of New Issues 632

The Costs of Going Public: A Case Study 634

20.8 The Rights Puzzle 640

20.10 Shelf Registration 643

20.11 Issuing Long-Term Debt 644

Mini Case: East Coast Yachts Goes Public 650

and Debt Capacity with Corporate Taxes 659

Present Value of Riskless Cash Flows 659 Optimal Debt Level and Riskless Cash Flows 660

21.6 NPV Analysis of the Lease-versus-Buy Decision 661

21.7 Debt Displacement and Lease Valuation 662

The Basic Concept of Debt Displacement 662 Optimal Debt Level in the Xomox Example 663

21.8 Does Leasing Ever Pay? The Base Case 665 21.9 Reasons for Leasing 666

21.10 Some Unanswered Questions 670

Are the Uses of Leases

Why Are Leases Offered by Both Manufacturers and Third-Party Lessors? 670 Why Are Some Assets Leased More

Mini Case: The Decision to Lease or Buy at

Appendix 21A: APV Approach to Leasing 676

PART VI Options, Futures, and Corporate Finance

22.7 Valuing Options 686

The Factors Determining Call Option Values 688

A Quick Discussion of Factors Determining

22.8 An Option Pricing Formula 691

22.9 Stocks and Bonds as Options 699

The Firm Expressed In Terms of

The Firm Expressed in Terms of Put Options 701

22.10 Options and Corporate Decisions:

Some Applications 704

22.11 Investment in Real Projects

Mini Case: Clissold Industries Options 720

Chapter 23

Options and Corporate Finance:

23.1 Executive Stock Options 722

23.4 Shutdown and Reopening Decisions 735

The Abandonment and Opening Decisions 736

Mini Case: Exotic Cuisines’

Chapter 24

24.2 The Difference between Warrants

and Call Options 747

How the Firm Can Hurt Warrant Holders 750

Black–Scholes Model 750 24.4 Convertible Bonds 752 24.5 The Value of Convertible Bonds 752

Mini Case: S&S Air’s Convertible Bond 765

Chapter 25

25.1 Derivatives, Hedging, and Risk 767 25.2 Forward Contracts 768 25.3 Futures Contracts 769

25.5 Interest Rate Futures Contracts 775

25.6 Duration Hedging 782

The Case of Two Bonds with the Same

25.8 Actual Use of Derivatives 792

Mini Case: Williamson Mortgage, Inc 798

Chapter 26

26.1 Tracing Cash and Net Working Capital 800

26.2 The Operating Cycle

and the Cash Cycle 801

Defining the Operating and Cash Cycles 802

The Operating Cycle and the

Calculating the Operating and Cash Cycles 804

26.3 Some Aspects of Short-Term

26.5 The Short-Term Financial Plan 816

Mini Case: Keafer Manufacturing Working

Chapter 27

27.1 Reasons for Holding Cash 829

The Speculative and Precautionary Motives 829

Cash Management versus Liquidity Management 830

27.2 Understanding Float 831

Electronic Data Interchange and Check 21:

27.3 Cash Collection and Concentration 837

Accelerating Collections: An Example 840

27.4 Managing Cash Disbursements 842

27.5 Investing Idle Cash 844

Characteristics of Short-Term Securities 845 Some Different Types of Money Market Securities 845

Mini Case: Cash Management at

Appendix 27A: Determining the

Appendix 27B: Adjustable Rate Preferred

Stock, Auction Rate Preferred Stock, and Floating-Rate Certificates of Deposit 850

Chapter 28

28.1 Credit and Receivables 851

28.2 Terms of the Sale 853

28.3 Analyzing Credit Policy 857

28.4 Optimal Credit Policy 859

28.8 Inventory Management Techniques 867

Mini Case: Credit Policy at Braam Industries 879

Appendix 28A: More about Credit

PART VIII Special Topics

Chapter 29

Mergers, Acquisitions, and Divestitures 880

29.1 The Basic Forms of Acquisitions 880

29.4 Two Financial Side Effects of

29.5 A Cost to Stockholders

from Reduction in Risk 892

How Can Shareholders Reduce Their Losses

29.6 The NPV of a Merger 894

29.7 Friendly versus Hostile Takeovers 898

The Managers versus the Stockholders 906

29.10 The Tax Forms of Acquisitions 908

29.11 Accounting for Acquisitions 910

29.12 Going Private and Leveraged Buyouts 911

Mini Case: The Birdie Golf—Hybrid Golf Merger 921

Chapter 30

30.1 What Is Financial Distress? 923 30.2 What Happens in Financial Distress? 925 30.3 Bankruptcy Liquidation

The Z-Score Model 935

31.3 Purchasing Power Parity 946

31.4 Interest Rate Parity, Unbiased Forward Rates, and the International Fisher Effect 949

31.5 International Capital Budgeting 953

Method 1: The Home Currency Approach 954 Method 2: The Foreign Currency Approach 955

The Cost of Capital for International Firms 956

31.6 Exchange Rate Risk 956

Summary and Conclusions 960

Mini Case: East Coast Yachts Goes

No Contents (syllabus) Corporate Finance (11th

Chapter 12: An Alternative View of Risk and Return – The Arbitrage Pricing Theory

2

Chapter 2: The effect of financial leverage on

risk and return

2.1 Operating leverage

2.2 Financial leverage

2.3 Total leverage

2.4 Overview of risk

2.5 Business and financial risks

Chapter 13: Risk, Cost of Capital and Valuation

3

Chapter 3: Sources of Funds

3.1 Overview

3.2 Financial market and Funding Instruments

3.3 Asset Structure and Funding Strategy

Chapter 15: Long-term Financing

Chapter 20: Raising Capital Chapter 21: Leasing

4

Chapter 4: Capital structure

4.1 Financial structure and capital structure

4.2 The relationship between capital structure

and firm value

4.3.1 Optimal capital structure theory

No Contents (syllabus) Corporate Finance (11th

Edition)

6

Chapter 6: Financial analysis

6.1 Overview of Financial Analysis

6.2 Content of Financial Analysis

6.3.1 Financial Indicators

6.3.2 The Dupont Identity

Chapter 3: Financial Statements Analysis and Financial Models

7

Chapter 7: Financial planning

7.1 Definition and role of financial planning

7.2 Classification of financial planning

7.3 Establishment of financial planning

7.4 What can planning accomplish?

7.5 Short-term financial planning

7.6 Long-term financial planning

Chapter 2: Financial Statements and Cash Flow

Chapter 15: Long-term Financing

Chapter 20: Raising Capital Chapter 21: Leasing

Chapter 26: Short – Term Finance and Planning Chapter 27: Cash Management Chapter 28: Credit and

Inventory Management

With the S&P 500 Index returning about 14 percent and

the NASDAQ Composite Index up about 13 percent in

2014, stock market performance overall was very good

In particular, investors in outpatient diagnostic imaging

services company RadNet, Inc., had to be happy about the

411 percent gain in that stock, and investors in

biopharma-ceutical company Achillon Pharmabiopharma-ceuticals had to feel pretty

good following that company’s 269 percent gain Of course,

not all stocks increased in value during the year Stock in

Transocean Ltd fell 63 percent during the year, and stock in Avon Products dropped 44 percent.

These examples show that there were tremendous potential profits to be made during 2014, but there was also the risk of losing money—and lots of it So what should you,

as a stock market investor, expect when you invest your own money? In this chapter, we study more than eight decades of market history to find out.

Risk and Return

LESSONS FROM MARKET HISTORY

10

Returns DOLLAR RETURNS

Suppose the Video Concept Company has several thousand shares of stock outstanding and you are a shareholder Further suppose that you purchased some of the shares of stock

in the company at the beginning of the year; it is now year-end and you want to figure out how well you have done on your investment The return you get on an investment in stocks, like that in bonds or any other investment, comes in two forms

As the owner of stock in the Video Concept Company, you are a part owner of the company If the company is profitable, it generally could distribute some of its profits to the shareholders Therefore, as the owner of shares of stock, you could receive some cash,

called a dividend, during the year This cash is the income component of your return In addition to the dividends, the other part of your return is the capital gain—or, if it is negative, the capital loss (negative capital gain)—on the investment.

For example, suppose we are considering the cash flows of the investment in Figure 10.1, showing that you purchased 100 shares of stock at the beginning of the year

at a price of $37 per share Your total investment, then, was:

10.1

How did the market

do today? Find out at

finance.yahoo.com.

coverage online

Excel Master

PART III: RISK

Suppose that over the year the stock paid a dividend of $1.85 per share During the year, then, you received income of:

Div 5 $1.85 3 100 5 $185Suppose, finally, that at the end of the year the market price of the stock is $40.33 per share Because the stock increased in price, you had a capital gain of:

Gain 5 ($40.33 2 $37) 3 100 5 $333The capital gain, like the dividend, is part of the return that shareholders require to main-tain their investment in the Video Concept Company Of course, if the price of Video Concept stock had dropped in value to, say, $34.78, you would have recorded this capital loss:

Loss 5 ($34.78 2 $37) 3 100 5 −$222

The total dollar return on your investment is the sum of the dividend income and the

capital gain or loss on the investment:

Total dollar return 5 Dividend income 1 Capital gain (or loss)

(From now on we will refer to capital losses as negative capital gains and not distinguish

them.) In our first example, the total dollar return is given by:

Total dollar return 5 $185 1 $333 5 $518Notice that if you sold the stock at the end of the year, your total amount of cash would be the initial investment plus the total dollar return In the preceding example you would have:

Total cash if stock is sold 5 Initial investment 1 Total dollar return

Figure 10.1

Dollar Returns

0 Time

Initial investment

Inflows

Outflows

The answer to the first question is a strong yes, and the answer to the second question is an equally strong no The capital gain is every bit as much a part of your return as the dividend, and you should certainly count it as part of your total return

That you have decided to hold onto the stock and not sell, or realize the gain or the

loss, in no way changes the fact that, if you wanted, you could get the cash value of the stock After all, you could always sell the stock at year-end and immediately buy

it back The total amount of cash you would have at year-end would be the $518 gain plus your initial investment of $3,700 You would not lose this return when you bought back 100 shares of stock In fact, you would be in exactly the same position as if you had not sold the stock (assuming, of course, that there are no tax consequences and no brokerage commissions from selling the stock)

PERCENTAGE RETURNS

It is more convenient to summarize the information about returns in percentage terms than

in dollars because the percentages apply to any amount invested The question we want to

answer is this: How much return do we get for each dollar invested? To find this out, let t stand for the year we are looking at, let P t be the price of the stock at the beginning of the year, and let Divt11 be the dividend paid on the stock during the year Consider the cash flows in Figure 10.2

In our example, the price at the beginning of the year was $37 per share and the dend paid during the year on each share was $1.85 Hence, the percentage income return,

divi-sometimes called the dividend yield, is:

Dividend yield 5 Divt11yP t

Dividends paid 1 Change in market

at end of period value over period

Beginning market value

1 1 Percentage return 5

Dividends paid 1  Market value

at end of period at end of period Beginning market value

The capital gain (or loss) is the change in the price of the stock divided by the initial

price Letting P t11 be the price of the stock at year-end, we can compute the capital gain

From now on, we will refer to returns in percentage terms

To give a more concrete example, stock in Keurig Green Mountain (GMCR), of coffee making by the cup fame, began 2014 at $75.54 per share Keurig Green Mountain paid divi-dends of $1.00 during 2014, and the stock price at the end of the year was $132.40 What was the return on GMCR for the year? For practice, see if you agree that the answer is 76.60 percent

Of course, negative returns occur as well For example, again in 2014, GameStop’s stock price

at the end of the year was $33.80 per share, and dividends of $1.32 were paid The stock began the year at $49.26 per share Verify that the loss was 28.70 percent for the year

EXAMPLE

10.1 Calculating Returns Suppose a stock begins the year with a price of $25 per share and ends with

a price of $35 per share During the year, it paid a $2 dividend per share What are its dividend yield, its capital gains yield, and its total return for the year? We can imagine the cash flows in Figure 10.3.

share (P1 ) 1

Holding Period Returns

A famous set of studies dealing with rates of return on common stocks, bonds, and

Treasury bills is found in Ibbotson SBBI 2015 Classic U.S Yearbook.1 It presents by-year historical rates of return for the following five important types of financial instru-ments in the United States:

year-1 Large-company common stocks: This common stock portfolio is based on the

Standard & Poor’s (S&P) Composite Index At present the S&P Composite includes

500 of the largest (in terms of market value) companies in the United States

2 Small-company common stocks: This is a portfolio corresponding to the bottom

fifth of stocks traded on the New York Stock Exchange in which stocks are ranked

by market value (i.e., the price of the stock multiplied by the number of shares standing)

out-3 Long-term corporate bonds: This is a portfolio of high-quality corporate bonds with

20-year maturities

4 Long-term U.S government bonds: This is based on U.S government bonds with

maturities of 20 years

5 U.S Treasury bills: This is based on Treasury bills with a one-month maturity.

None of the returns are adjusted for taxes or transaction costs In addition to the by-year returns on financial instruments, the year-to-year change in the consumer price index is computed This is a basic measure of inflation We can calculate year-by-year real returns by subtracting annual inflation

year-Before looking closely at the different portfolio returns, we graphically present the returns and risks available from U.S capital markets in the 89-year period from 1926 to

2014 Figure 10.4 shows the growth of $1 invested at the beginning of 1926 Notice that the vertical axis is logarithmic, so that equal distances measure the same percentage change The figure shows that if $1 was invested in large-company common stocks and all dividends were reinvested, the dollar would have grown to $5,316.85 by the end of 2014 The biggest growth was in the small stock portfolio If $1 was invested in small stocks in 1926, the investment would have grown to $27,419.32 However, when you look carefully at Figure 10.4, you can see great variability in the returns on small stocks, especially in the earlier part of the period

A dollar in long-term government bonds was very stable as compared with a dollar in mon stocks Figures 10.5 to 10.8 plot each year-to-year percentage return as a vertical bar drawn from the horizontal axis for large-company common stocks, small-company stocks, long-term bonds and Treasury bills, and inflation, respectively

com-10.2

coverage online

Excel Master

1Ibbotson SBBI 2015 Classic Yearbook (Chicago: Morningstar).

Thus, the stock’s dividend yield, its capital gains yield, and its total return are 8 percent, 40 percent, and 48 percent, respectively.

Suppose you had $5,000 invested The total dollar return you would have received on an ment in the stock is $5,000 3 48 5 $2,400 If you know the total dollar return on the stock, you do not need to know how many shares you would have had to purchase to figure out how much money you would have made on the $5,000 investment You just use the total dollar return.

invest-Figure 10.4 gives the growth of a dollar investment in the stock market from 1926 through 2014 In other words, it shows what the value of the investment would have been

if the dollar had been left in the stock market and if each year the dividends from the

previous year had been reinvested in more stock If R t is the return in year t (expressed

in decimals), the value you would have at the end of year T is the product of 1 plus the

return in each of the years:

Value 5 (1 1 R1) 3 (1 1 R2) 3 · · · 3 (1 1 R t ) 3 · · · 3 (1 1 R T)For example, if the returns were 11 percent, 25 percent, and 9 percent in a three-year period, an investment of $1 at the beginning of the period would be worth:

a three-year holding period return Table 10.1 gives the annual returns each year for

selected investments from 1926 to 2014 From this table, you can determine holding period returns for any combination of years

Redrawn from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates to the work by Roger G Ibbotson and Rex A Sinquefield (Chicago:

Morningstar) All rights reserved.

Treasury bills

Long-term government bonds

Large-company stocks Small-company stocks

Figure 10.4 Wealth Indexes of Investments in the U.S Capital Markets (Year-End 1925 5 $1.00)

Figure 10.5 Year-by-Year Total Returns on Large-Company Common Stocks

240

220 0

1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

20 40 60

260

Redrawn from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates work by Roger G Ibbotson and Rex A Sinquefield

(Chicago: Morningstar) All rights reserved.

Figure 10.6 Year-by-Year Total Returns on Small-Company Stocks

2100

250 0

1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

50 100 150 200

Redrawn from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates work by Roger G Ibbotson and Rex A Sinquefield

(Chicago: Morningstar) All rights reserved.

Redrawn from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates to the work by Roger G Ibbotson and Rex A Sinquefield

(Chicago: Morningstar) All rights reserved.

Figure 10.7 Year-by-Year Total Returns on Bonds and U.S Treasury Bills

220

210 0

1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

10 20 30 40 50

22 0

1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

2 4 6 8 10 12 14 16

Year Large-Company Stocks

Long-Term Government Bonds

U.S

Treasury Bills Consumer Price Index

1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944

11.14%

37.13 43.31

−8.91

−25.26

−43.86

−8.85 52.88

−2.34 47.22 32.80

−35.26 33.20

−.91

−10.08

−11.77 21.07 25.76 19.69

7.90%

10.36

−1.37 5.23 5.80

−8.04 14.11 31 12.98 5.88 8.22

−.13 6.26 5.71 10.34

−8.66 2.67 2.50 2.88

3.30%

3.15 4.05 4.47 2.27 1.15 88 52 27 17 17 27 06 04 04 14 34 38 38

21.12%

22.26 21.16 58 26.40 29.32 210.27 76 1.52 2.99 1.45 2.86 22.78 00 71 9.93 9.03 2.96 2.30

1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

5 10 15 20

Figure 10.8 Year-by-Year Inflation

Redrawn from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates to the work by Roger G Ibbotson and Rex A

Sinquefield (Chicago: Morningstar) All rights reserved.

Table 10.1

Year-by-Year Total Returns, 1926–2014

(continued ) Year

Large-Company Stocks

Long-Term Government Bonds

U.S

Treasury Bills Consumer Price Index

1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989

36.46%

−8.18 5.24 5.10 18.06 30.58 24.55 18.50

− 1.10 52.40 31.43 6.63 210.85 43.34 11.90 48 26.81

−8.78 22.69 16.36 12.36

−10.10 23.94 11.00

−8.47 3.94 14.30 18.99

−14.69

−26.47 37.23 23.93

−7.16 6.57 18.61 32.50

−4.92 21.55 22.56 6.27 31.73 18.67 5.25 16.61 31.69

5.17%

4.07

−1.15 2.10 7.02

−1.44

−3.53 1.82

−.88 7.89

−1.03

−3.14 5.25

−6.70

−1.35 7.74 3.02 4.63 1.37 4.43 1.40

−1.61

−6.38 5.33

−7.45 12.24 12.67 9.15

−12.66

−3.28 4.67 18.34 2.31

−2.07

−2.76

−5.91

−.16 49.99

−2.11 16.53 39.03 32.51

−8.09 8.71 22.15

.38%

.38 62 1.06 1.12 1.22 1.56 1.75 1.87 93 1.80 2.66 3.28 1.71 3.48 2.81 2.40 2.82 3.23 3.62 4.06 4.94 4.39 5.49 6.90 6.50 4.36 4.23 7.29 7.99 5.87 5.07 5.45 7.64 10.56 12.10 14.60 10.94 8.99 9.90 7.71 6.09 5.88 6.94 8.44

2.25%

18.13 8.84 2.99 22.07 5.93 6.00 75 75 2.74 37 2.99 2.90 1.76 1.73 1.36 67 1.33 1.64 97 1.92 3.46 3.04 4.72 6.20 5.57 3.27 3.41 8.71 12.34 6.94 4.86 6.70 9.02 13.29 12.52 8.92 3.83 3.79 3.95 3.80 1.10 4.43 4.42 4.65

Return Statistics

The history of U.S capital market returns is too complicated to be handled in its undigested form To use the history, we must first find some manageable ways of describing it, dra-matically condensing the detailed data into a few simple statements

This is where two important numbers summarizing the history come in The first and most natural number is some single measure that best describes the past annual returns on the stock market In other words, what is our best estimate of the return that an investor could

have realized in a particular year over the 1926 to 2014 period? This is the average return.

Figure 10.9 plots the histogram of the yearly stock market returns given in Table 10.1

This plot is the frequency distribution of the numbers The height of the graph gives the

number of sample observations in the range on the horizontal axis

Given a frequency distribution like that in Figure 10.9, we can calculate the average

or mean of the distribution To compute the average of the distribution, we add up all of

the values and divide by the total (T ) number (89 in our case because we have 89 years

of data) The bar over the R is used to represent the mean, and the formula is the ordinary

formula for calculating an average:

Year Large-Company Stocks

Long-Term Government Bonds

U.S

Treasury Bills Consumer Price Index

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

−3.10%

30.46 7.62 10.08 1.32 37.58 22.96 33.36 28.58 21.04

−9.10

−11.89

−22.10 28.68 10.88 4.91 15.79 5.49

−37.00 26.46 15.06 2.11 16.00 32.39 13.7

5.44%

20.04 8.09 22.32

−11.46 37.28

−2.59 17.70 19.22

−12.76 22.16 5.30 14.08 1.62 10.34 10.35 28 10.85 19.24 225.61 7.73 35.75 1.80 –14.69 12.9

7.69%

5.43 3.48 3.03 4.39 5.61 5.14 5.19 4.86 4.80 5.98 3.33 1.61 1.03 1.43 3.30 4.97 4.52 1.24 15 14 06 08 05 03

6.11%

3.06 2.90 2.75 2.67 2.54 3.32 1.70 1.61 2.68 3.39 1.55 2.38 1.88 3.26 3.42 2.54 4.08 09 2.72 1.50 2.96 1.74 1.50 8

SOURCE: Global Financial Data, www.globalfinancialdata.com, copyright 2015 en.m.wikipedia.org, Treasury.gov, bls.gov.

Table 10.1

Year-by-Year Total

Returns, 1926–2014

(concluded )

10.2 Calculating Average Returns Suppose the returns on common stock from 1926 to 1929 are

.1370, 3580, 4514, and −.0888, respectively The average, or mean return over these four years is:

1958 1935 1928

2001 1973 1966 1957 1941

2009 2003 1999 1998 1996 1983 1982 1976 1967 1963 1961 1951 1943 1942

2000 1990 1981 1977 1969 1962 1953 1946 1940 1939 1934 1932 1929

2011 2007 2005 1994 1993 1992 1987 1984 1978 1970 1960 1956 1948 1947

2014 2012 2010 2006 2004 1988 1986 1979 1972 1971 1968 1965 1964 1959 1952 1949 1944 1926

2013 1997 1995 1991 1989 1985 1980 1975 1955 1950 1945 1938 1936 1927 Percent

100 110 120 130 Small-Company Stocks

1937 1929

2013 2010 1995 1983 1980 1968 1961 1950 1938 1928

2011 2007 2000 1998 1987 1984 1966 1960 1953 1948 1941 1940 1932

2005 1994 1986 1972 1956 1952 1951 1947 1939 1926

150 140

1945 1933

2008 1973 1930

1967 1943

2002 1974 1970 1962 1957 1946

1991 1979 1965 1942 1935

2009 2001 1999 1997 1993 1992 1988 1985 1982 1978 1977 1964 1963 1955 1934 1927

1976 1975 1944

2003 1958 1954 1936

1990 1969

Percent

160 170

Redrawn from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates to the work by Roger G Ibbotson and Rex A Sinquefield (Chicago: Morningstar)

All rights reserved.

Figure 10.9 Histogram of Returns on U.S Common Stocks, 1926–2014

Average Stock Returns and Risk-Free Returns

Now that we have computed the average return on the U.S stock market, it seems sensible

to compare it with the returns on other securities The most obvious comparison is with the low-variability returns in the U.S government bond market These are free of most of the volatility we see in the stock market

An interesting comparison, then, is between the virtually risk-free return on T-bills and the very risky return on common stocks This difference between risky returns and

risk-free returns is often called the excess return on the risky asset Of course, in any

par-ticular year the excess return might be positive or negative

Table 10.2 shows the average stock return, bond return, T-bill return, and inflation rate for the period from 1926 through 2014 From this we can derive average excess returns

The average excess return from large-company common stocks relative to T-bills for the

on common stocks is called the historical equity risk premium because it is the additional

return from bearing risk

One of the most significant observations of stock market data is this long-term excess

of the stock return over the risk-free return An investor for this period was rewarded for investment in the stock market with an extra, or excess, return over what would have been achieved by simply investing in T-bills

Why was there such a reward? Does it mean that it never pays to invest in T-bills, and that someone who invested in them instead of in the stock market needs a course

in finance? A complete answer to these questions lies at the heart of modern finance

However, part of the answer can be found in the variability of the various types of ments There are many years when an investment in T-bills achieves higher returns than

invest-an investment in large common stocks Also, we note that the returns from invest-an investment

in common stocks are frequently negative, whereas an investment in T-bills has never produced a negative return So, we now turn our attention to measuring the variability of returns and an introductory discussion of risk

Risk Statistics

The second number that we use to characterize the distribution of returns is a measure

of the risk in returns There is no universally agreed-upon definition of risk One way

to think about the risk of returns on common stock is in terms of how spread out the frequency distribution in Figure 10.9 is The spread, or dispersion, of a distribution is a measure of how much a particular return can deviate from the mean return If the distri-bution is very spread out, the returns that will occur are very uncertain By contrast, a distribution whose returns are all within a few percentage points of each other is tight, and the returns are less uncertain The measures of risk we will discuss are variance and standard deviation

VARIANCE

The variance and its square root, the standard deviation, are the most common measures

of variability, or dispersion We will use Var and s2 to denote the variance, and we will use SD and s to represent the standard deviation s is, of course, the Greek letter sigma

10.4

10.5

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Excel Master

Table 10.2 Total Annual Returns, U.S Securities Markets, 1926–2014

Series

Arithmetic Mean (%)

Standard Deviation (%) Distribution (%)

Small-company stocks *

Intermediate-term government bonds

* The 1933 small-company stock total return was 142.9 percent.

SOURCE: Modified from Stocks, Bonds, Bills and Inflation: 2015 Yearbook,™ annual updates to the work by Roger G Ibbotson and Rex A Sinquefield (Chicago:

Morningstar) All rights reserved.

EXAMPLE

10.3 Volatility Suppose the returns on common stocks are (in decimals) 1370, 3580, 4514, and

2.0888, respectively The variance of this sample is computed as follows:

Var = 1 _ T − 1 [(R1 − R ) 2 + (R2 − R )2 + (R3 − R )2 + (R4 − R )2 ] 0582 = 1 3 [(.1370 − 2144)2 + (.3580 − 2144) 2

+ (.4514 − 2144) 2 + (−.0888 − 2144) 2 ]

SD = √ _.0582 = 2412 or 24.12%

This formula in Example 10.3 tells us just what to do: Take the T individual returns (R1, R2, …) and subtract the average return R −, square the result, and add them up Finally,

this total must be divided by the number of returns less one (T 2 1) The standard deviation

is always just the square root of the variance

Using the stock returns for the 89-year period from 1926 through 2014 in this formula, the resulting standard deviation of large-company stock returns is 20.1 percent The standard devia-tion is the standard statistical measure of the spread of a sample, and it will be the measure we use most of the time Its interpretation is facilitated by a discussion of the normal distribution

Standard deviations are widely reported for mutual funds For example, the Fidelity Magellan Fund is a large mutual fund How volatile is it? To find out, we went to www.morningstar.com, entered the ticker symbol FMAGX, and hit the “Ratings &

Risk” link Here is what we found:

Over the last three years, the standard deviation of the returns on the Fidelity Magellan Fund was 10.41 percent When you consider the average stock has a standard deviation of about 50 percent, this seems like a low number But the Magellan Fund is a relatively well-diversified portfolio, so this is an illustration of the power of diversifica-tion, a subject we will discuss in detail later The return given is the average return; so over the last three years, investors in the Magellan Fund earned an 18.77 percent return

per year Also under the Volatility Measures section, you will see the Sharpe ratio The

Sharpe ratio is calculated as the risk premium of the asset divided by the standard tion As such, it is a measure of return to the level of risk taken (as measured by standard deviation) The “beta” for the Fidelity Magellan Fund is 1.07 We will have more to say about this number—lots more—in the next chapter

devia-NORMAL DISTRIBUTION AND ITS IMPLICATIONS FOR STANDARD DEVIATION

A large enough sample drawn from a normal distribution looks like the bell-shaped curve

drawn in Figure 10.10 As you can see, this distribution is symmetric about its mean, not

skewed, and has a much cleaner shape than the actual distribution of yearly returns drawn in Figure 10.9 Of course, if we had been able to observe stock market returns for 1,000 years,

we might have filled in a lot of the jumps and jerks in Figure 10.9 and had a smoother curve

In classical statistics, the normal distribution plays a central role, and the standard deviation is the usual way to represent the spread of a normal distribution For the normal distribution, the probability of having a return that is above or below the mean by a certain amount depends only on the standard deviation For example, the probability of having a

EXAMPLE

10.4 Sharpe Ratio The Sharpe ratio is the average equity risk premium over a period of time divided

by the standard deviation From 1926 to 2014 the average risk premium (relative to Treasury bills) for large-company stocks was 8.6 percent while the standard deviation was 20.1 percent The Sharpe ratio of this sample is computed as:

Sharpe ratio 5 8.6%y20.1% 5 428 The Sharpe ratio is sometimes referred to as the reward-to-risk ratio where the reward is the average excess return and the risk is the standard deviation.

Figure 10.10

The Normal Distribution

There is a 95.44 percent probability that a return will be within two standard deviations of the mean

In this example, there is a 95.44 percent probability that a yearly return will be between 228.1 percent and 52.3 percent.

Finally, there is a 99.74 percent probability that a return will be within three standard deviations

of the mean In this example, there is a 99.74 percent probability that a yearly return will be between 248.2 percent and 72.4 percent.

return that is within one standard deviation of the mean of the distribution is approximately 68, or 2y3, and the probability of having a return that is within two standard deviations of the mean is approximately 95.

The 20.1 percent standard deviation we found for stock returns from 1926 through 2014 can now be interpreted in the following way: If stock returns are roughly normally distributed, the probability that a yearly return will fall within 20.1 percent of the mean of 12.1 percent will

be approximately 2y3 That is, about 2y3 of the yearly returns will be between 28.0 percent and 32.2 percent (Note that 28.0 5 12.1 2 20.1 and 32.2 5 12.1 1 20.1.) The probability that the return in any year will fall within two standard deviations is about 95 That is, about

95 percent of yearly returns will be between 228.1 percent and 52.3 percent

More on Average Returns

Thus far in this chapter we have looked closely at simple average returns But there is another way of computing an average return The fact that average returns are calculated two different ways leads to some confusion, so our goal in this section is to explain the two approaches and also the circumstances under which each is most appropriate

ARITHMETIC VERSUS GEOMETRIC AVERAGES

Let’s start with a simple example Suppose you buy a particular stock for $100

Unfortunately, the first year you own it, it falls to $50 The second year you own it, it rises back to $100, leaving you where you started (no dividends were paid)

What was your average return on this investment? Common sense seems to say that your average return must be exactly zero because you started with $100 and ended with $100 But if we calculate the returns year-by-year, we see that you lost 50 percent the first year (you lost half of your money) The second year, you made 100 percent (you doubled your money) Your average return over the two years was thus (250 percent 1

100 percent)y2 5 25 percent!

So which is correct, 0 percent or 25 percent? The answer is that both are correct; they

just answer different questions The 0 percent is called the geometric average return

The 25 percent is called the arithmetic average return The geometric average return

answers the question, “What was your average compound return per year over a

particu-lar period?” The arithmetic average return answers the question, “What was your return

in an average year over a particular period?”

Notice that in previous sections, the average returns we calculated were all arithmetic averages, so we already know how to calculate them What we need to do now is (1) learn how to calculate geometric averages and (2) learn the circumstances under which average

is more meaningful than the other

CALCULATING GEOMETRIC AVERAGE RETURNS

First, to illustrate how we calculate a geometric average return, suppose a particular ment had annual returns of 10 percent, 12 percent, 3 percent, and 29 percent over the last four years The geometric average return over this four-year period is calculated as (1.10 3 1.12 3 1.03 3 91)1/4 2 1 5 3.66 percent In contrast, the average arithmetic return

invest-we have been calculating is (.10 1 12 1 03 2 09)y4 5 4.0 percent

In general, if we have T years of returns, the geometric average return over these T

years is calculated using this formula:

Geometric average return 5 [(1 1 R1) 3 (1 1 R2) 3 ? ? ? 3 (1 1 R T) ] 1/T 2 1 (10.1)

10.6

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Excel Master

This formula tells us that four steps are required:

1 Take each of the T annual returns R1, R2, , R T and add 1 to each (after converting them to decimals)

2 Multiply all the numbers from step 1 together

3 Take the result from step 2 and raise it to the power of 1yT

4 Finally, subtract 1 from the result of step 3 The result is the geometric average return

EXAMPLE

10.5 Calculating the Geometric Average Return Calculate the geometric average return for S&P

500 large-cap stocks for a five-year period using the numbers given here.

First convert percentages to decimal returns, add 1, and then calculate their product:

S&P 500 Returns Product

13.75%

35.70 45.08

Notice that the number 1.5291 is what our investment is worth after five years if we started with a $1 investment The geometric average return is then calculated as:

Geometric average return 5 1.5291 1/5 2 1 5 0887, or 8.87%

Thus, the geometric average return is about 8.87 percent in this example Here is a tip: If you are using a financial calculator, you can put $1 in as the present value, $1.5291 as the future value, and

5 as the number of periods Then solve for the unknown rate You should get the same answer

we did.

You may have noticed in our examples thus far that the geometric average returns seem to be smaller It turns out that this will always be true (as long as the returns are not all identical, in which case the two “averages” would be the same) To illustrate, Table 10.3 shows the arithmetic averages and standard deviations from Table 10.2, along with the geometric average returns

As shown in Table 10.3, the geometric averages are all smaller, but the magnitude

of the difference varies quite a bit The reason is that the difference is greater for more volatile investments In fact, there is a useful approximation Assuming all the numbers are expressed in decimals (as opposed to percentages), the geometric average return is approximately equal to the arithmetic average return minus half the variance For example, looking at the large-company stocks, the arithmetic average is 12.1 and the standard devia-tion is 201, implying that the variance is 040 The approximate geometric average is thus 121 2 _ .0402 5 100 (510.0%), which is very close to the actual value

ARITHMETIC AVERAGE RETURN OR GEOMETRIC AVERAGE RETURN?

When we look at historical returns, the difference between the geometric and arithmetic age returns isn’t too hard to understand To put it slightly differently, the geometric average tells you what you actually earned per year on average, compounded annually The arithmetic average tells you what you earned in a typical year and is an unbiased estimate of the true mean of the distribution The geometric average is very useful in describing the actual histori-cal investment experience The arithmetic average is useful in making estimates of the future.2

aver-The U.S Equity Risk Premium: Historical and International Perspectives

So far, in this chapter, we have studied the United States in the period from 1926 to 2014

As we have discussed, the historical U.S stock market risk premium has been substantial Of course, anytime we use the past to predict the future, there is a danger that the past period isn’t representative of what the future will hold Perhaps U.S investors got lucky over this period

Arithmetic Mean

Standard Deviation

SOURCE: Ibbotson SBBI 2015 Classic Yearbook.

EXAMPLE

10.6 More Geometric Averages Take a look back at Figure 10.4 There we showed the value of a

$1 investment after 89 years Use the value for the large-company stock investment to check the geometric average in Table 10.3.

In Figure 10.4, the large-company investment grew to $5,316.85 over 89 years The geometric average return is thus:

Geometric average return 5 $5,316.85 1/89 2 1 5 101, or 10.1%

This 10.1 percent is the value shown in Table 10.3 For practice, check some of the other numbers

in Table 10.3 the same way.

2 Another way of thinking about estimating an investment’s return over a particular future horizon is to recall from your statistics class that the arithmetic average is a “sample” mean As such, it provides an unbiased estimate of the underlying true mean To use the arithmetic average to estimate the future returns, we must make sure the historical returns are measured using the same interval as the future forecasting period For example, we could use yearly (annual) returns to estimate next year’s return The arithmetic average would be a good basis for forecasting the next two-year returns if two-year holding period returns were used We also must be confident that the past distribution of returns is the same as that of the future.

and earned particularly large returns Data from earlier years for the United States is available, though it is not of the same quality With that caveat in mind, researchers have tracked returns back to 1802, and the U.S equity risk premium in the pre-1926 era was smaller Using the U.S return data from 1802, the historical equity risk premium was 5.4 percent.3

Also, we have not looked at other major countries Actually, more than half of the value of tradable stock is not in the United States From Table 10.4, we can see that while the total world stock market capitalization was more than $34 trillion in 2014, only about

58 percent was in the United States Thanks to Dimson, Marsh, and Staunton, data from earlier periods and other countries are now available to help us take a closer look at equity risk premiums Table 10.5 and Figure 10.11 show the historical stock market risk premi-ums for 17 countries around the world in the period from 1900 to 2010 Looking at the numbers, the U.S historical equity risk premium is the 7th highest at 7.2 percent (which differs from our earlier estimate because of the different time periods examined) The over-all world average risk premium is 6.9 percent It seems clear that U.S investors did well, but not exceptionally so relative to many other countries The top-performing countries according to the Sharpe ratio were the United States, Australia, South Africa, and France, while the worst performers were Belgium, Norway, and Denmark Germany, Japan, and Italy might make an interesting case study because they have the highest stock returns over this period (despite World Wars I and II), but also the highest risk

So what is a good estimate of the U.S equity risk premium going forward? Unfortunately, nobody can know for sure what investors expect in the future If history is a guide, the expected U.S equity risk premium could be 7.2 percent based upon estimates from 1900–2010 We should also be mindful that the average world equity risk premium was 6.9 percent over this same period On the other hand, the more recent periods (1926–2014) suggest higher estimates

of the U.S equity risk premium, and earlier periods going back to 1802 suggest lower estimates

3 Jeremy J Siegel has estimated the U.S equity risk premium with data from 1802 As can be seen in the following table, from

1802 to 2011 the historical equity risk premium was 5.4 percent.

Average Returns 1802–2011 (%)

Common stock Treasury bills Equity risk premium

9.6 4.2 5.4

Adopted and updated from J Siegel, Stocks for the Long Run, 4th ed (New York: McGraw-Hill, 2008).

Table 10.5

Annualized Equity

Risk Premiums and

Sharpe Ratios for

17 Countries,

1900–2010

Country

Historical Equity Risk Premiums (%)

(1)

Standard Deviation (%) (2)

The Sharpe Ratio (1) / (2)

Australia Belgium Canada Denmark France Germany*

Ireland Italy Japan Netherlands Norway South Africa Spain Sweden Switzerland United Kingdom United States

Average

8.3%

5.5 5.6 4.6 8.7 9.8 5.3 9.8 9.0 6.5 5.9 8.3 5.4 6.6 5.1 6.0 7.2

6.9

17.6%

24.7 17.2 20.5 24.5 31.8 21.5 32.0 27.7 22.8 26.5 22.1 21.9 22.1 18.9 19.9 19.8

23.0

.47 22 33 22 36 31 25 31 32 29 22 30 25 30 27 30 36

.30

*Germany omits 1922–1923.

SOURCE: Elroy Dimson, Paul Marsh, and Michael Staunton, Credit Suisse Global Investment Returns Sourcebook, 2011, published

by Credit Suisse Research Institute 2011 The Dimson-Marsh-Staunton data set is distributed by Morningstar, Inc.

Denmark Switzerland Ireland SpainBelgiumCanadaNorway UK

Netherlands Sweden

US

South Africa Australia France JapanGermany Italy

0 2 4

8 6

10 12

SOURCE: Elroy Dimson, Paul Marsh, and Michael Staunton, “The Worldwide Equity Premium: A Smaller Puzzle,” in Handbook of

the Equity Risk Premium, Rajnish Mehra, ed (Elsevier, 2007) Updates by the authors.