SSRN_ID223797_code000918140

Demystifying the Use of Beta in the Determination of the Cost of Capital and an Illustration of Its Use in Lazard’s

Valuation of Conrail

Samuel C. Thompson, Jr.*

I. INTRODUCTION....................................................................................................................242 A. Net Present Value ........................................................................................................242 B. Cost of Capital .............................................................................................................243 C. Beta and the Capital Asset Pricing Model..............................................................244 D. Purpose of the Article .................................................................................................246

II. INTRODUCTION TO BETA...................................................................................................247 A. Systematic and Unsystematic Risk ............................................................................247 B. Beta: A Measure of Systematic Risk .........................................................................249 C. Background Information on Return on the Market and on Four Securities .....249

III. DERIVATION OF BETA THROUGH THE USE OF COVARIANCE AND VARIANCE..........252 A. Introduction ..................................................................................................................252 B. Expected Return ...........................................................................................................252 C. Variance ........................................................................................................................254 D. Covariance ...................................................................................................................258

1. Introduction ............................................................................................................258 2. Standard Deviation................................................................................................258 3. Correlation Coefficient .........................................................................................260 4. Computation of Covariance.................................................................................264

E. The Determination of Beta .........................................................................................264

IV. DERIVATION OF BETA THROUGH REGRESSION ANALYSIS ..........................................266 A. Introduction ..................................................................................................................266 B. Scatter Diagrams .........................................................................................................266 C. The Method of Least Squares ....................................................................................268 D. Normal Equations for Least Squares Line..............................................................268 E. Shortcut Method for Computing Slope-Beta...........................................................275 F. Coefficient of Determination .....................................................................................277

V. DERIVATION OF BETA THROUGH THE USE OF EXCEL’S LINEAR REGRESSION FUNCTON .............................................................................................................................282

* Professor and Director, Center for the Study of Mergers and Acquisitions, University of Miami School of Law, and Tax Policy Advisor (on behalf of the United States Treasury Department), Ministry of Finance, Republic of South Africa. The author would like to thank Tarek Sayed, a 1998 graduate of the University of Miami School of Law and a student in the Graduate Program in Taxation at the University of Miami School of Law, and Gheri Hicks, a third-year student at the University of Miami School of Law, for their assistance in the preparation of this art icle.

242 The Journal of Corporation Law [Winter

VI. SUMMARY OF BETA RESULTS AND INTRODUCTION TO ADJUSTED BETAS................283 A. Summary ........................................................................................................................283 B. Adjusted Betas..............................................................................................................284

VII. ASSET AND EQUITY BETAS AND THE LEVERING AND UNLEVERING FORMULA........284 A. Introduction ..................................................................................................................284 B. The Levering and Unlevering Formula ...................................................................288

VIII. RETURNING TO THE CAPITAL ASSET PRICING MODEL .................................................290

IX. LAZARD’S USE OF BETA, CAPM, AND THE FORMULA FOR LEVERING AND DELEVERING BETA IN COMPUTING CONTRAIL’S WACC AND PRESENT VALUE.....293 A. Introduction ..................................................................................................................293 B. Computation of Levered Beta ....................................................................................294 C. Conversion of Levered Beta to Unlevered Beta .....................................................294 D. Recomputation of Levered Beta for Various Debt to Equity Ratios...................296 E. Use of CAPM to Compute Cost of Levered Equity ................................................297 F. Computation of Conrail’s Weighted Average Cost of Capital.............................298 G. Use of WACC in Determining the Present Value of Conrail’s Free Cash

Flow...............................................................................................................................300

X. SUMMARY OF LAZARD’S DCF VALUATION APPROACH..............................................302

XI. CONCLUSION.......................................................................................................................303

I. INTRODUCTION

A. Net Present Value

One of the principal methods of valuation in mergers and acquisitions and other capital budgeting contexts is the net present value (NPV) technique, which can be broken down into four steps.1 First, this technique requires an estimate of the free cash flows and the terminal value expected to be generated by the target firm (target) or other investment.2 Free cash flow is the actual cash flow, not accounting earnings, that the target can expect to generate. Thus, depreciation and other noncash deductions used in 1. See Samuel C. Thompson, Jr., A Lawyer’s Guide to Modern Valuation Techniques in Mergers and Acquisitions, 21 J. CORP . L. 457, 528-30 (1996). The present article is an extension and elaboration on many of the concepts discussed in Part VI of the earlier article. See id. at 501-20. See generally RICHARD A. BREALEY & STEWART C. MYERS, PRINCIPLES OF CORPORATE FINANCE (5th ed. 1996); EUGENE F. BRIGHAM & LOUIS C. GAPENSKI, FINANCIAL MANAGEMENT: THEORY AND PRACTICE (8th ed. 1997); JOHN Y. CAMPBELL ET AL., THE ECONOMETRICS OF FINANCIAL MARKETS (1997); T OM COPELAND ET AL., VALUATION: MEASURING AND MANAGING THE VALUE OF COMPANIES (2d ed. 1994); DAMODAR N. GUJARATI , BASIC ECONOMETRICS (3d ed. 1995); BURTON G. MALKIEL, A RANDOM WALK DOWN WALL STREET (6th ed. 1996); ROBERT C. RADCLIFFE, INVESTMENT: CONCEPTS, ANALYSIS, AND STRATEGY (5th ed. 1997); ALFRED RAPPAPORT, CREATING SHAREHOLDER VALUE (1986); MILTON L. ROCK ET AL., T HE MERGERS AND ACQUISITIONS HANDBOOK (2d ed. 1994); STEPHEN A. ROSS ET AL., FUNDAMENTALS OF CORPORATE FINANCE (2d ed. 1993); and J. FRED WESTON ET AL., T AKEOVERS, RESTRUCTURING, AND CORPORATE GOVERNANCE (2d ed. 1997). Several of the statistical concepts discussed here are explored in the following statistics books: DAVID K. HILDEBRAND & R. LYMAN OTT, BASIC STATISTICAL IDEAS FOR MANAGERS (3d ed. 1996) and MURRAY R. SPIEGEL, SCHAUM’S OUTLINE OF T HEORY AND PROBLEMS OF STATISTICS (2d ed. 1991). 2. See Thompson, supra note 1, at 481-500. The discussion here will focus on the acquisition of a target.

2000] Demystifying the Use of Beta 243

computing net income are added back, and capital expenditures and other nondeductible cash expenditures are deducted in computing free cash flow. The terminal value is the cash expected to be realized from the sale of the target after an assumed holding period, such as five or ten years.

Second, the cost of capital, which is the return required for an investment in the target, must be determined. Third, the estimated free cash flows and terminal value are discounted to present value at the applicable cost of capital, or discounted rate, to arrive at the aggregate present value of these cash flows.3 These first three steps are generally referred to as the discounted cash flow (DCF) method.

Fourth, the NPV of the transaction is determined by subtracting the cost of the target from the aggregate present value of the target’s cash flows and terminal value. If this amount is positive, the investment has a positive NPV. In such a case, because the investment will offer a return that exceeds the cost of capital of the investment, the investment should generally be made.4 Thus, the key variables used in the NPV technique are the estimate of the free cash flows and terminal value and the determination of the cost of capital.

In an article entitled The Valuation of Cash Flow Forecasts: An Empirical Analysis, Professors Steven Kaplan and Richard Ruback find that “discounted cash flow valuation methods provide reliable estimates of market value [of management buyouts and leveraged recapitalizations].”5 This analysis provides strong theoretical support for the proposition that the DCF and related NPV techniques are important valuation tools in merger and acquisition transactions.

B. Cost of Capital

Depending on the particular context, the cost of capital may be the weighted average cost of capital (WACC) appropriate for the target firm,6 the cost of equity assuming an unleveraged target firm, or the cost of equity assuming a leveraged target firm. For example, Professor Alfred Rappaport generally recommends the use of WACC in determining the cost of capital.7 WACC automatically takes account of the financing decision by placing the appropriate weight on the cost of equity and the after-tax cost of the debt. Because the interest on debt is tax deductible for federal income tax purposes,8 the use of debt will cause the WACC to be lower than the cost of equity. This approach is referred to here as the WACC DCF approach.

Richard A. Brealey and Stewart C. Myers, on the other hand, generally suggest that the financing decision be separated from the investment decision by discounting the cash

3. Id. at 469-81. 4. Id. at 474-75. Even if an investment produces a negative NPV, the investment may be undertaken if, for example, an option to make a follow-on investment that could result in an aggregate positive NPV is embedded in the initial investment decision. The impact of “real options” like this are discussed in BREALEY & MYERS, supra note 1, at 589-610. See also Keeping All Options Open, ECONOMIST, Aug. 14, 1999, at 62. 5. Steven N. Kaplan & Richard S. Ruback, The Valuation of Cash Flow Forecasts: An Empirical Analysis, 50 J. FIN. 1054, 1091 (1995). 6. Thompson, supra note 1, at 522-25. 7. RAPPAPORT, supra note 1, at 55; see also Thompson, supra note 1, at 522-25. 8. I.R.C. § 163 (1986).


flows and terminal value at the cost of unlevered equity for the target.9 The impact of the use of debt is separately accounted for by discounting the expected tax savings from any interest deduction at the applicable cost of equity capital and adding the present value of the tax saving to the NPV to determine the adjusted NPV or APV.10 This is referred to here as the APV DCF approach. Professors Kaplan and Ruback use what they refer to as the “Compressed APV” technique in conducting their DCF valuation study.11

Under both the WACC and APV approaches, interest is not deducted when computing the free cash flows, because the effect of the use of interest is accounted for with a lower cost of capital with WACC or by separately discounting the tax savings from the interest deduction with APV. This is referred to as the all equity method of computing free cash flow, that is, unlevered free cash flow.12 The WACC and APV methods produce a valuation of the target’s assets; the target’s outstanding debt is then subtracted to determine the value of the target’s equity.

In following a third approach, Eugene F. Brigham and Louis C. Gapenski suggest that a target’s free cash flows, after deducting interest, plus terminal value, be discounted at the target’s leveraged cost of capital.13 Because this approach is valuing the equity, debt is not subtracted from the valuation as is the case with WACC and APV. This approach is referred to here as the equity DCF approach. Properly applied, these three methods should produce similar estimates of the fair market value of the target’s equity. All of these methods require an estimate of the cost of equity: the cost of levered equity under WACC and the Brigham and Gapenski technique, and the cost of unlevered equity under the Brealey and Myers APV technique.

Investment banking firms generally seem to use the WACC DCF approach and, therefore, use the all equity assumption in creating a cash flow statement, thus focusing on an unlevered free cash flow. This is illustrated in the cash flow statement used by the investment banking firm of Lazard Frères & Co., LLC in its valuation of Conrail,14 which was the target of an acquisition by competing acquirors CSX and Norfolk Southern.15 It is also illustrated in the 1989 acquisition of Warner Communications, Inc. by Time, Inc. In that case, the investment bankers for both Time and Warner used unlevered free cash flow statements in their DCF analyses.16

C. Beta and the Capital Asset Pricing Model

One of the principal ways of determining the cost of equity is the capital asset

9. BREALEY & MYERS, supra note 1, at 120. 10. Id. at 528-31; see also Thompson, supra note 1, at 521-22. 11. See Kaplan & Rubak, supra note 5, at 1062. 12. See generally BREALEY & MYERS, supra note 1, at 120. 13. BRIGHAM & GAPENSKI, supra note 1, at 1087-89. Brigham and Gapinski refer to the method as the “equity residual method to valu[ing] the target firm, because the estimated net cash flows are a residual which belongs solely to the acquiring firm’s shareholders.” Id. at 1089. 14. See infra Appendix B. 15. This transaction is discussed in Norfolk Southern Corp. v. Conrail Inc., Civ. Act. No. 96-7197 (E.D. Pa. 1997) (Vanartsdalen, J., ruling from the bench), reproduced in SAMUEL C. THOMPSON, JR., BUSINESS PLANNING FOR MERGERS AND ACQUISITIONS (1997). 16. See Thompson, supra note 1, at apps. C-3, D-5.


pricing model (CAPM).17 CAPM provides that the cost of equity capital (re) for a target is determined by a formula that takes into account the following variables: (1) the risk-free rate of return (rf); (2) the market risk premium (rm-rf), which is the difference between the expected return on the market (rm) the risk-free rate of return (rf); and (3) the beta (β) for the target’s stock. Beta measures the relationship between the movements of the returns on a stock with the movements of the equity market.18 The formula is as follows:19

Expected Rate of Return (re) = [Risk-free Rate of Return (rf)] + Beta (β) x [Market Risk Premium (rm-rf)]

re = rf + β (rm - rf)

The risk-free rate of return is either the T-bill rate, as suggested by Brealey & Myers20 and John Y. Campbell, Andrew W. Lo, and A. Craig MacKinlay,21 or the T-bond rate, as suggested by Brigham and Gapenski.22 The market risk premium is the difference between the expected return on a market portfolio of common stocks (rm) (such as the S&P 500)23 and the risk-free rate (rf). Brealey and Myers report that historically the market risk premium when the T-bill rate is used as the risk-free rate has been 8.4% per year,24 and Tom Copeland, Tim Koller, and Jack Murrin report that if the ten year T-bond rate is taken as the risk-free rate, the market risk premium is 5 to 6%.25 Thus, if the T-bill rate is used as the risk-free rate, the market risk premium will be higher than if the T-bond rate is used.26

While the risk-free rate and the market risk premium are determined by observing market factors, the beta is specific to the target. As will be seen, the beta for the market 17. This model may be used for determining the cost of equity capital for purposes of applying the economic valued added (EVA) concept. See, e.g., AL EHRBAR, EVA: THE REAL KEY TO CREATING WEALTH 178-79 (1998). “EVA . . . is a measure of corporate performance that . . . includ[es] a charge against profit for the cost of all the capital a company employs.” Id. at 1. 18. BREALEY & MYERS, supra note 1, at 160-61. “Beta measures the amount that investors expect the stock price to change for each additional 1 percent change in the market.” Id. at 166. 19. Thompson, supra note 1, at 502 (citing RICHARD A. BREALEY & STEWART C. MYERS, PRINCIPLES OF CORPORATE FINANCE 190-97 (4th ed. 1991)). 20. BREALEY & MYERS, supra note 1, at 147; see also Thompson, supra note 1, at 510-11. 21. CAMPBELL ET AL., supra note 1, at 184. 22. BRIGHAM & GAPENSKI, supra note 1, at 170. Brigham and Gapenski say that the risk-free rate in a

CAPM analysis can be proxied by either a long-term rate (the T -bond rate) or a short-term rate (the T -bill rate). Traditionally the T-bill rate was used, but in recent years, there has been a movement toward the use of the T-bond rate because there is a closer relationship between T-bond yields and stocks than between T-bill yields and stocks.

Id. at 173 n.13. 23. CAMPBELL ET AL., supra note 1, at 184 (suggesting use of the S&P 500 as the proxy for the market portfolio). 24. BREALEY & MYERS, supra note 1, at 146. 25. COPELAND ET AL., supra note 1, at 260. 26. For example, Ibbotson Associates found the average risk premium of stocks, which had an average return of 12.5%, over T -bonds to be 7.4%, and the average risk premium of stocks over T-bills to be 8.8%. BRIGHAM AND GAPENSKI, supra note 1, at 351.


portfolio is 1 and, consequently, if the target has a risk profile that is the same as the market, the target will also have a beta of 1. In such case, the required cost of capital for the target will be the same as the market’s rate of return. This can be seen as follows. Assume that the market rate of return is 12%, the risk-free rate is 5%, and the target’s beta is 1. Consequently, the required rate of return is 12%, computed as follows:

r = rf + β (rm - rf) r = 5% + 1(12% - 5%) r = 5% + 1(7%) r = 12%

If the target is more risky than the market portfolio, the target’s beta will be higher than 1, and the target’s required rate of return will be higher than the return on the market. If the target is less risky than the market portfolio, the target’s beta will be lower than 1, and the target’s required rate of return will be lower than the return on the market. Brigham and Gapenski report that betas for a single stock may not be stable over time;27 that is, the beta for the period from 1985 to 1990 may differ from the beta for the period 1990 to 1995. They report that betas of a portfolio of ten or more randomly selected stocks are reasonably stable.28 They go on to state: “The conclusion that follows from the beta stability studies is that the CAPM is a better concept for structuring investment portfolios than it is for estimating the cost of capital for individual securities.”29 This conclusion could mean that in determining the beta for a particular company, it may be prudent to consider the betas for similar companies. On this point, Copeland, Koller, and Murrin say that “Because measurement errors tend to cancel out, industry averages are more stable than company betas.”30

D. Purpose of the Article

The purpose of this Article is to first discuss the derivation of beta and then to illustrate how beta and CAPM were used by the investment banking firm of Lazard Frères & Co., LLC (“Lazard”) in determining the cost of capital for Conrail and in valuing Conrail’s free cash flows. Although the validity of beta and CAPM have been challenged,31 Brigham and Gapenski say that CAPM and beta are “useful conceptual tool[s].”32 Also, Brealey and Myers say that CAPM is the “best-known model of risk and return,”33 and that CAPM is “widely used by large corporations to estimate the discount 27. BRIGHAM & GAPENSKI, supra note 1, at 214. 28. Id. 29. Id.; see also BREALEY & MYERS, supra note 1, at 209 (“Potentially large estimate errors [can arise] when you estimate betas of individual stocks from a limited sample of data. Fortunately these errors tend to cancel when you estimate betas of portfolios.”). 30. COPELAND ET AL., supra note 1, at 264. 31. See, e.g., Eugene F. Fama & Kenneth R. French, The Cross-Section of Expected Stock Returns, 47 J. FIN. 427 (1992); Thompson, supra note 1, at 517-20; BRIGHAM & GAPENSKI, supra note 1, at 213-16; CAMPBELL ET AL., supra note 1, at 211-17. 32. BRIGHAM & GAPENSKI, supra note 1, at 216. 33. BREALEY & MYERS, supra note 1, at 194.


rate,” that is, the cost of capital.34 Further, in evaluating the utility of beta, Burton G. Malkiel concludes that the “reports of beta’s total demise are, in my judgment, premature.”35

CAPM and beta are used by investment bankers in preparing, inter alia, merger and acquisition valuations. This is illustrated in Appendix A, which contains the computation by Lazard of the cost of equity and WACC of Conrail, which, at the time, was the target of both CSX and Norfolk Southern. This Appendix illustrates how Lazard first determined Conrail’s beta and then used CAPM with both leveraged and unleveraged betas to determine Conrail’s cost of equity under various debt-equity assumptions. Lazard then used this range of costs of equities to determine a range of WACCs for Conrail. This Appendix also illustrates the application of the formula for moving from levered to unlevered betas and then back to levered betas. Appendix B shows how Lazard used Conrail’s WACC as determined in Appendix A to determine the present value of Conrail’s free cash flows and terminal value.

Parts II through VI address the derivation of beta, Part VII discusses the formula for moving between unlevered and levered betas, and Part VIII elaborates on the use of beta in CAPM. Part IX first analyzes Lazard’s use in Appendix A of CAPM and beta in computing Conrail’s WACC and then discusses Lazard’s use in Appendix B of this cost of capital in determining the value of Conrail’s cash flows. Finally, Part X provides a brief summary of Lazard’s WACC DCF valuation of Conrail.

II. INTRODUCTION TO BETA36

A. Systematic and Unsystematic Risk

There are two types of risk inherent in investments: systematic risk and unsystematic risk. Risk in either context means that future returns are uncertain .37 Systematic risk is a function of broad macroeconomic conditions that affect the prices of all assets.38 Thus, there is systematic risk in holding any asset. Conversely, unsystematic risk is a function of the characteristics associated with a particular asset as opposed to broad market factors.39 For example, risk factors such as the rate of growth of GNP or interest rate levels have a systematic effect on all firms .40 Comparatively, risk factors such as possible changes in consumer tastes, new product developments, and changes in prices of raw materials affect particular firms and, therefore, are referred to as unsystematic risks.41

Diversification can reduce or eliminate unsystematic risk by balancing the losing stocks with winning stocks; such risk is often referred to as diversifiable risk, unique risk, residual risk, or specific risk.42 The specific risk of holding stock in a particular company

34. Id. at 206; see also WESTON ET AL., supra note 1, at 189-90. 35. MALKIEL, supra note 1, at 271. 36. Part II.A-B is largely excerpted from the discussion in Thompson, supra note 1, at 502-04. 37. Id. at 502 (citing BREALEY & MYERS, supra note 19, at 149; RONALD J. GILSON & BERNARD S. BLACK , T HE LAW AND FINANCE OF CORPORATE ACQUISITIONS 81-100 (2d ed. 1995)). 38. Thompson, supra note 1, at 502 (citing BREALEY & MYERS, supra note 19, at 137). 39. Thompson, supra note 1, at 502-03 (citing BREALEY & MYERS, supra note 19, at 137). 40. Thompson, supra note 1, at 503 (citing GILSON & BLACK , supra note 37, at 108). 41. Thompson, supra note 1, at 503 (citing GILSON & BLACK , supra note 37, at 107). 42. Thompson, supra note 1, at 503 (citing BREALEY & MYERS, supra note 19, at 137).


can be diversified away by investing in stocks of other companies, possibly in other industries . Diversification cannot eliminate systematic risk, however. Systematic risk is often referred to as undiversifiable risk or market risk.43 Because market risk results from broad macroeconomic factors that threaten most businesses, stocks have a tendency to move in the same direction.44

Brealey and Myers elaborate on the concepts of unique (or unsystematic) risk and market (or systematic) risk:

If you only have a single stock, unique risk is very important, but once you have a portfolio of 20 or more stocks, diversification has done the bulk of its work. For a reasonably well-diversified portfolio, only market risk matters . Therefore, the predominant source of uncertainty for a diversified investor is that the market will rise or plummet, carrying the investor’s portfolio with it.45

The following graph illustrates total risk as a function of both systematic and unsystematic risk and the reduction of unsystematic risk through diversification.46

This graph illustrates that the greater the diversification (represented on the horizontal axis), the less the total risk (measured on the vertical axis). Unsystematic risk is reduced by diversification; systematic risk is not. Diversification reduces unsystematic risk because the prices of different stocks move differently, the price movements are less than perfectly correlated, and the decline in the prices of some stocks are offset by increases in the prices of others.47 Brealey and Myers explain that “With more securities, and therefore better diversification, portfolio risk declines until all unique risk is 43. Thompson, supra note 1, at 503 (citing BREALEY & MYERS, supra note 19, at 137). 44. Thompson, supra note 1, at 503 (citing BREALEY & MYERS, supra note 19, at 139). 45. Thompson, supra note 1, at 503 (citing BREALEY & MYERS, supra note 19, at 139). 46. Thompson, supra note 1, at 503 (citing BREALEY & MYERS, supra note 19, at 139 and GILSON & BLACK , supra note 37, at 89-93). 47. Thompson, supra note 1, at 504 (citing BREALEY & MYERS, supra note 19, at 137).


eliminated and only the bedrock of market risk remains.”48 Thus, the market portfolio has only systematic ris k.

B. Beta: A Measure of Systematic Risk

Beta is the measure of the sensitivity of a security’s return to market movements.49 Thus, it is a measure of the systematic risk of a security.50 Beta indicates what the likely move for the particular stock will be for a given move in the market, that is, how sensitive a particular stock is to a move in the market. For a stock with a beta of 1, a 10% market rise would, on average, lead to a 10% rise in the price of the stock. For a stock with a beta of 1.5, a market rise of 10% would lead on average to a 15% rise in the price of the stock. Thus, beta “measures the marginal contribution of a stock to the risk of the market portfolio.”51 Beta “gauges the tendency of the return on a security to move in parallel with the return of the stock market as a whole.”52

The beta for a particular stock is calculated by running a linear regression between past returns for that stock and past returns of a market index, such as the Standard and Poor’s 500,53 or by dividing (1) the covariance between the returns on the particular stock and the returns on the market index by (2) the variance of the market portfolio. Part III shows how beta is derived through the use of the covariance and variance concepts, Part IV derives beta through the use of regression analysis, and Part V shows how beta can be derived through the use of the linear regression function in the Excel program.

Betas are generally computed on the basis of past data (i.e., ex post). Techniques exist, however, for estimating ex ante, or future, betas.54 Although many financial organizations, such as Merrill Lynch,55 publish beta estimates for many companies, and beta estimates are provided through a service offered by Bloomberg,56 one of the purposes of this Article is to give the reader an intuitive understanding of the statistical concepts behind beta. This will be of particular assistance in those circumstances when the analyst needs to construct his or her own beta estimate.

C. Background Information on Return on the Market and on Four Securities

To illustrate the computation of beta, the following returns are assumed to have been obtained for a particular period (which in this case is assumed to be eight years) for the market portfolio and for three different stocks (stocks 1, 2, and 3). This data is shown

48. Thompson, supra note 1, at 504 (quoting BREALEY & MYERS, supra note 19, at 144). 49. Thompson, supra note 1, at 504 (citing BREALEY & MYERS, supra note 19, at 143). 50. Thompson, supra note 1, at 504 (citing ALAN C. SHAPIRO, MODERN CORPORATE FINANCE 118 (1990) and GILSON & BLACK , supra note 37, at 94-98). 51. Thompson, supra note 1, at 504 (quoting BREALEY & MYERS, supra note 19, at 163). 52. Thompson, supra note 1, at 504 (quoting David W. Mullins, Jr., Does the Capital Asset Pricing Model Work?, HARV. BUS. REV., Jan.-Feb. 1982, at 105, 108). 53. Thompson, supra note 1, at 504 (citing RAPPAPORT, supra note 1, at 58). 54. Thompson, supra note 1, at 504 (citing RAPPAPORT, supra note 1, at 58-59). 55. MERRILL LYNCH , SECURITY RISK EVALUATION (Jan. 2000). The use of the Merrill Lynch beta book is discussed in BREALEY & MYERS, supra note 1, at 207-12. 56. Data provided by Bloomberg includes “complete descriptive information, volume, and output from Bloomberg analytics such as beta . . .” The Bloomberg Service (visited Feb. 18, 2000) <http://www.bloomberg. com/products/prod_terminal02.html>.


below in frequency distributions, which show the frequency with which each return occurred. This is referred to as a discrete distribution, because there are a finite number of possible returns.57 Also, Tables A, B, and C show the probability (i.e., a probability distribution) of the particular outcome (i.e., return), which is determined by analyzing historical data concerning the market’s or stock’s possible returns and the frequency of those returns. To simplify matters, there are only three returns for each stock. It is assumed that the highest, middle, and lowest returns for each stock are perfectly correlated with the highest, middle, and lowest returns for the market; for example, when the market produces its highest return, each stock produces its highest return, and so on.58 This assumption is important in computing the correlation coefficient in Part III.D.3 and in performing the regression analysis in Part IV.

Table A Frequency and Probability Distribution

for the Market Portfolio [1] [2] [3] Market's

Actual Returns Frequency of Returns Over 8-Year Period

Probability of Outcome

(1) 15% 3 40% (2) 10% 2 20% (3) -0- 3 40% TOTALS 8 100%

57. RADCLIFFE , supra note 1, at 181. 58. The data in Tables A through C were derived from the following chart:

RETURNS OF: YEAR MARKET STOCK 1

AT&T STOCK 2 GROWTH

STOCK

STOCK 3 UTILITY

STOCK 1. 1991 15% 15% 20% 10% 2. 1992 0% 0% -5% 5% 3. 1993 15% 15% 20% 10% 4. 1994 10% 10% 15% 7% 5. 1995 0% 0% -5% 5% 6. 1996 10% 10% 15% 7% 7. 1997 0% 0% -5% 5% 8. 1998 15% 15% 20% 10%


Table B Frequency and Probability Distribution

for Stock 1 (AT&T) [1] [2] [3] Stock's



(1) 15% 3 40% (2) 10% 2 20% (3) -0- 3 40% TOTALS 8 100%

Table C Frequency and Probability

Distribution for Stock 2 (Growth Stock)

[1] [2] [3] Stock's



(1) 20% 3 40% (2) 15% 2 20% (3) -5% 3 40% TOTALS 8 100%

Table D

Frequency and Probability Distribution for Stock 3

(Utility Stock) [1] [2] [3] Stock's



(1) 10% 3 40% (2) 7% 2 20% (3) 5% 3 40% TOTALS 8 100%

The above data is dramatically simplified for illustrative purposes. Brigham and

Gapenski report that most analysts use five years of data in computing betas, and they seem to suggest the use of weekly returns for a five-year period, which would produce a


sample size of 260 (e.g., 52 weeks x 5 years = 260).59 Campbell, Lo, and Mackinlay suggest the use of five years of monthly returns,60 which would produce a sample size of 60 (e.g., 12 months x 5 years = 60).

III. DERIVATION OF BETA THROUGH THE USE OF COVARIANCE AND VARIANCE

A. Introduction

As indicated, the beta for a particular stock is determined by dividing: (1) the covariance between (a) the returns on the particular stock, and (b) the returns on the market portfolio, by (2) the variance of the market portfolio . Covariance here is a measure of the “degree of parallelism” between the returns on a particular stock and the returns on the market portfolio.61 The variance of the market portfolio indicates the potential for deviation of the market’s actual return from the market’s expected return, and the variance for a particular security indicates the potential for deviation of the security’s actual return from the security’s expected return. Thus, variance and the associated concept of standard deviation are measures of financial risk, that is, “the chance that expected security returns will not materialize and, in particular, that the securities you hold will fall in price.”62

B. Expected Return

The market’s expected return (i.e., more generally the expected value) is the profit expected from the market as a rate of return for a specified period. A security’s expected return is the return expected for the security for a specified period. The expected return is computed by multiplying each of the actual returns by the probability that such return would have occurred and summing the results. Thus, the starting point in computing the expected return is to construct a probability distribution of returns as set forth in Part II.C. Brigham and Gapenski explain, “If we multiply each possible outcome by its probability of occurrence and then sum these products, . . . we have a weighted average of outcomes. The weights are the probabilities, and the weighted average is the expected rate of return . . . .”63

The expected rate of return E(R) is expressed in the following equation:

E(R) = P1r1 + P2r2 + P3r3 . . . + Pnrn

= ∑=

n

i

iirP1

In the above equations, P1 through Pn are the probabilities associated with returns r1 through rn.64 59. BRIGHAM & GAPENSKI, supra note 1, at 354. 60. CAMPBELL ET AL., supra note 1, at 184. 61. MALKIEL, supra note 1, at 237. 62. Id. at 229. 63. BRIGHAM & GAPENSKI, supra note 1, at 148. 64. More generally, this equation can be written as:


The computations of the expected returns for the market portfolio and the three stocks set forth in Part II.C are set forth below:

Table E Market's Expected Return

[1] [2] [3] Market's

Actual Returns Probability of Outcome

(PDF)

Expected Return [1] x [2]

(1) 15% 40% 6% (2) 10% 20% 2% (3) -0- 40% 0% TOTAL

Expected Return = 8%

Table F Expected Return of Stock 1

(AT&T) [1] [2] [3] Stock's


(PDF)


(1) 15% 40% 6% (2) 10% 20% 2% (3) -0- 40% 0% TOTAL


∑ ∑=

=n

i

ii xxfX1

)()(

Here Σ(X) is the expected value of the discrete random variable (X), f(xi) is the probability distribution function (PDF) of X, and xi represents the possible values of X. GUJARATI , supra note 1, at 763. The PDF (e.g., the probability that the discrete random variable X has the value xi) sums to 1, thus accounting for all possible outcomes of X. Id. at 758. This is demonstrated on Tables E through H. The equation above for the expected value of X is sometimes written in the following simple form:

∑ ∑=x

xxfX )()(

Id. at 763.


Table G Expected Return of Stock 2

(Growth Stock) [1] [2] [3] Stock's


(PDF)


(1) 20% 40% 8% (2) 15% 20% 3% (3) -5% 40% -2% TOTAL


Table H Expected Return of Stock 3

(Utility Stock) [1] [2] [3] Market's

Actual Returns


(PDF)


(1) 10% 40% 4% (2) 7% 20% 1.4% (3) 5% 40% 2% TOTAL

Expected Return = 7.4%

In summary, the expected return is the weighted average return, which is found by multiplying each possible return by the corresponding probability and summing the results. Brigham and Gapenski explain that “The tighter, or more peaked, the probability distribution, the more likely it is that the actual [return] will be close to the expected [return] . . . .”65

C. Variance

The variance of a random variable such as the returns on the market or a security “is the probability-weighted average of squared deviations from the mean (expected value).”66 The formula for the variance of a random variable (Y) can be written as follows:67

Var(Y) = σy2 = Σ(Y - µy)2 Py(Y)

65. BRIGHAM & GAPENSKI, supra note 1, at 149. 66. HILDEBRAND & OTT, supra note 1, at 118; see also GUJARATI , supra note 1, at 765. 67. HILDEBRAND & OTT, supra note 1, at 118.


In the formula, µy is the expected value of Y, which can be written E(Y) , while Py(Y) is the probability distribution for each value of y. David K. Hildebrand and R. Lyman Ott explain that in calculating Var(Y), one takes “each value y, subtract[s] the expected value µy = E(Y), square[s] the result, multipl[ies] by the probability Py(y), and sum[s].”68 In computing variance, the expected return of the market or of a particular security is subtracted from the actual return for a particular period or periods. This difference is then multiplied by the probability of such an occurrence. The resulting amount is then squared to get the variance. The formula for the variance of the market (σµ

2) can be set out in words as follows:69

∑=

−=n

i

ii PRER1

22 )]([µσ

Thus, there is a three-step process in computing variance that can be illustrated as follows. (Note that the measure of variance is stated in the same units as the expected return, which in this case is a percentage.)70 The first step is to subtract the market’s or security’s expected return from the actual return to compute the deviation from the mean as follows:

Table I Market’s Deviation from the Mean

Actual Return for Particular Periods

Expected Returns Deviation from Mean

15% 8% 7 10% 8% 2 0% 8% -8

Table J

Stock 1's Deviation from the Mean Actual Return for Particular Periods


15% 8% 7 10% 8% 2 0% 8% -8

68. Id. at 119. 69. BREALEY & MYERS, supra note 1, at 149-51. 70. RADCLIFFE , supra note 1, at 183.


Table K Stock 2's Deviation from the Mean


Expected Returns Deviation From Mean

20% 9% 11 15% 9% 6 -5% 9% -14

Table L Stock 3's Deviation from the Mean



10% 7.4% 2.6 7% 7.4% -0.4 5% 7.4% -2.4

The second step is to square the deviations from the mean in order to obtain all

positive numbers, as follows:

Table M Market’s Squared Deviation

7 x 7 = 49 2 x 2 = 4

-8 x -8 = 64

Table N Stock 1’s Squared Deviation

7 x 7 = 49 2 x 2 = 4

-8 x -8 = 64

Table O Stock 2’s Squared Deviation

11 x 11 = 121 6 x 6 = 36

-14 x -14 = 196


Table P Stock 3’s Squared Deviation

2.6 x 2.6 = 6.76 -0.4 x -0.4 = 0.16 -2.4 x -2.4 = 5.76

The third step is to multiply the squared deviation by the probability of occurrence

of the particular actual return and sum to get the variance.

Table Q Market’s Variance

Squared Deviation x Probability = Variance 49 x .40 = 19.60 4 x .20 = 0.8 64 x .40 = 25.60 TOTAL 46 Variance

Table R Stock 1's Variance

Squared Deviation x Probability = Variance 49 x .40 = 19.60 4 x .20 = 0.8 64 x .40 = 25.60 TOTAL 46 Variance

Table S Stock 2's Variance

Squared Deviation x Probability = Variance 121 x .40 = 48.40 36 x .20 = 7.20 196 x .40 = 78.40

TOTAL 134 Variance

Table T Stock 3's Variance

Squared Deviation x Probability = Variance 6.76 x .40 = 2.70 .16 x .20 = .032 5.76 x .40 = 2.30

TOTAL 5.032 Variance


The market’s expected return and variance show the probability distribution associated with the stocks constituting the market.

D. Covariance

1. Introduction

Covariance is a measure of the way two random variables, such as the returns in the market and the returns on stock 1, move in relation to each other, that is, the way they “covary.”71 It is a measure of the way the returns on a particular stock move in relation to the returns on the market. The covariance between stock 1 and the market portfolio (µ) is determined by multiplying (1) the correlation coefficient72 (COR1µ) for stock 1 and the market by (2) the standard deviations (σ) of stock 1 and the market. The correlation coefficient measures the degree to which the movements of the prices of two variables are correlated, that is, move together. The standard deviation is the square root of the variance.

The covariance formula can be written as follows:

Covariance Between Stock 1 and the Market = σ1µ = COR1µ σ1σµ

This formula posits that the covariance between stock 1 and the market (σ1µ) is equal to the correlation coefficient between stock 1 and the market (COR1µ), multiplied by the standard deviations for stock 1 and the market (σ1 σµ). The covariance, between the returns on a particular stock and the returns on the market portfolio, measures the way the expected returns on the stock move in relation to the expected returns of the market.

If the covariance is positive, the returns of stock 1 and the market move in the same direction. If the covariance is negative, the returns of the stock and the market (the variables) move in opposite directions.

2. Standard Deviation

The standard deviation is the positive square root of the variance of a variable.73 The standard deviation expresses the dispersion of returns in units that can be more easily understood.74 Radcliffe explains that “the term standard can be thought of as standardizing squared return deviations back to a non squared value.”75

The standard deviation of the market is written as follows:

Standard Deviation (σm) of Variation (σm2) = )(ó VARIANCE 2

m

Continuing this illustration, the square root of the market’s variance of 46 is 6.78. Thus, the standard deviation is 6.78. This means that for normal distributions,76 68.27% 71. BREALEY & MYERS, supra note 1, at 157. 72. Id. 73. Id. 74. RADCLIFFE , supra note 1, at 183. 75. Id. 76. A normal distribution will produce a bell-shaped curve. Although “the pattern of historical returns


of the actual returns on the market are within one standard deviation (i.e., 6.78) on either side of the expected return; 95.45% of the actual returns are within two such standard deviations, and 99.73% of the actual returns are within three such standard deviations. The standard deviation for stocks 1, 2, and 3 are set out below:

Table U Standard Deviation of Stocks 1, 2, and 3

STOCK

VARIANCE VARIANCE

=

STANDARD DEVIATION

1 46 46 = 6.78 2 134 134 = 11.57 3 5.036 5.036 = 2.24

Brigham and Gapenski explain that “The smaller the standard deviation, the tighter

the probability distribution, and, accordingly, the lower the riskiness of the stock.”77 Thus, two stocks could have the same expected returns but different standard deviations.78 This can be illustrated as follows:

Although both stocks A and B have the same expected return, stock A has less risk as indicated by its smaller deviation from the expected return. Thus, when stock A has a smaller deviation from the expected return, it is said to dominate stock B.79

from individual securities has not usually been symmetric, the returns from well-diversified portfolios of stocks do seem to be distributed approximately symmetrically.” MALKIEL, supra note 1, at 231. 77. BRIGHAM & GAPENSKI, supra note 1, at 150-51. 78. RADCLIFFE , supra note 1, at 218. 79. Id.


In general, a stock with a higher expected return will also have a higher standard deviation. For example, as illustrated in the following table, stock 2 above has both a higher expected return and a higher standard deviation than stocks 1 and 3.

Table V

Comparison of Expected Returns and Standard Deviations Stock Expected Return Standard Deviation

1 8% 6.78% 2 9% 11.57% 3 7.4% 2.24%

Investors obviously would prefer a stock with the highest expected return and the lowest standard deviation, that is, the lowest risk of a return that would vary substantially from the expected return.

Brigham and Gapenski also report that in recent years “the standard deviation of a one-stock portfolio [(σ1)] (or an average stock), is approximately 35 percent. A portfolio consisting of all stocks, which is called the market portfolio, would have a standard deviation, σM, of about 18 percent . . . .”80 Thus, in Graph A in Part II.A, which illustrates systematic and nonsystematic risk, the total of the systematic and nonsystematic risk associated with the average one-stock portfolio has a standard deviation of about 35%, and the market portfolio, which has only systematic risk, has a standard deviation of 18%. Consequently, total risk is approximately equally divided between systematic and nonsystematic risk.

3. Correlation Coefficient

The correlation coefficient is a measure of the strength of the linear relationship between two random variables such as the return on the market (x) and the return on stock 1 (y). “The stronger the correlation, the better x [the independent or explanatory variable] predicts y [the dependent variable].”81 Correlation is closely related to the concept of regression, or estimation of a dependent variable (y) from an independent variable (x).82 Correlation determines the “degree of relationship between variables, which seeks to determine how well a linear or other equation describes or explains the relationship between variables.”83 If the dependent variable y tends to increase as the independent variable x increases the correlation is said to be positive or direct. On the other hand, if y tends to decrease as x increases, the correlation is said to be negative.84 Simple correlation or simple regression involves only two variables; multiple correlation and multiple regression involve more than two independent variables.85

80. BRIGHAM & GAPENSKI, supra note 1, at 163. 81. HILDEBRAND & OTT, supra note 1, at 457. 82. SPIEGEL, supra note 1, at 294. 83. Id. 84. Id. 85. Id.


This Part focuses only on correlation, while Part IV addresses the determination of beta from regression analysis and demonstrates how the correlation coefficient can be determined from the coefficient of determination.86 Beta is determined from one independent variable (i.e., market return) and, therefore, involves simple correlation or simple regression. Arbitrage Pricing Theory, which is an alternative to CAPM for determining the cost of capital, involves the use of multiple independent variables.87

If all values of the independent and dependent variables satisfy the correlation or regression equation exactly, the variables are said to be “perfectly correlated.” If there is no correlation between two variables, the variables are said to be “uncorrelated.”88

Mathematically the correlation coefficient, which is generally denoted rxy, is written as follows:

))(( 22 yx

xyrxy

ΣΣ

Σ=

In this equation, x = X - X and y = Y - Y .89 Thus, the formula for the correlation coefficient can be written more precisely as follows:

))-)()-(()()(

22

22

yyxx

yyxxrxy

ΣΣ

−−Σ=

This formula is also sometimes written as:90

SyySxxSxy

rxy

=

Here S means summation, that is Σ. The terms can be further defined as follows:91

∑∑ −−==i

ii yyxxSxyxy ))((

∑∑ −==i

i xxSxxx 22 )(

∑∑ −==i

i yySyyy 22 )(

86. See infra Part IV.F. 87. MALKIEL, supra note 1, at 272-74. 88. SPIEGEL, supra note 1, at 294. 89. Id. at 298. 90. HILDEBRAND & OTT, supra note 1, at 457. 91. Id. at 430-31, 458.


Σxy and Sxy are the sum of the x deviations times y deviations; Σx2 and Sxx are the sum of the x deviations times the x deviations, that is, the sum of the x deviations squared; and Σy2 or Syy are the sum of the y deviations squared.92 “This formula, which automatically gives the proper sign of r, [positive or negative], is called the product-moment formula and clearly shows the symmetry between x and y.”93

Using the formula, the correlation coefficients between the market and stocks 1, 2, and 3 can be computed as follows.94 First, tables showing the relationship between the independent variable, x (e.g., the return on the market), and the dependent variables, y (e.g., the returns on stocks 1, 2, and 3), are first constructed as follows:95

Table W

Relationship Between Returns on Market and Returns on 3 Stocks Relationship Between Return on the Market

(x) and Return on Stock 1 (y)

Relationship Between Return on the Market (x) and Return on Stock 2 (y)

Relationship Between Return on the Market (x) and Return on Stock 3

(y) Market Stock Market Stock Market Stock

15% 15% 15% 20% 15% 10% 10% 10% 10% 15% 10% 7% 0% 0% 0% -5% 0% 5%

Second, a table showing each element of the formula for computing the correlation

coefficient is prepared as follows for each stock. The first two columns in this table repeat the information from the above tables showing the relationship between the market’s return and the return for each of stock 1, 2, and 3. The third step is to plug into the formula for determining the correlation coefficient the information from the above correlation coefficient tables:

Table X Correlation Coefficient Table for Relationship Between Market and Stock 1

[1] [2] [3] [4] [5] [6] [7] x

Market Return

y

Stock 1 Return

x = x - x

x = 8%

y = y - y

y = 8%

x2

Squaring the amounts in [3]

xy

[3] x [4]

y2


15% 15% 7 7 49 49 49 10% 10% 2 2 4 4 4 0% 0% -8 -8 64 64 64

Σx2 = 117 Σxy = 117 Σy2 = 117

92. Id. 93. SPIEGEL, supra note 1, at 298. 94. Id. at 308. 95. Table W is constructed from the information on the returns for the eight-year period set out in note 58, supra .


Table Y

Correlation Coefficient Table for Relationship Between Market and Stock 2 [1] [2] [3] [4] [5] [6] [7] x

Market Return

y

Stock 2 Return

x = x - x

x = 8%

y = y - y

y = 9%

x2


xy

[3] x [4]

y2


15% 20% 7 11 49 77 121 10% 15% 2 6 4 12 36

0 -5% -8 -14 64 112 196 Σx2 = 117 Σxy = 201 Σy2 = 353

Table Z

Correlation Coefficient Table for Relationship Between Market and Stock 3 [1] [2] [3] [4] [5] [6] [7] x

Market Return

y

Stock 3 Return

x = x - x

x = 8%

y = y - y

y = 7.4%

x2


xy

[3] x [4]

y2


15% 10% 7 2.6 49 18.2 6.76 10% 7% 2 -0.4 4 -0.8 0.16 0% 5% -8 -2.4 64 19.2 5.76

Σx2 = 117 Σxy = 36.6 Σy2 = 12.68

The third step is to plug into the formula for determining the correlation coefficient the information from the above corellation coefficient tables:

Table AA

Correlation Coefficient Between Market and theThree Stocks Correlation Coefficient

Between Market and Stock 1 Correlation Coefficient

Between Market and Stock 2 Correlation Coefficient

Between Market and Stock 3

))(( 22 Yx

xyr

ΣΣ

Σ= ))(( 22 Yx

xyr

ΣΣ

Σ= ))(( 22 Yx

xyr

ΣΣ

Σ=

)117()117(117

×=r

)353()117(201

×=r

)68.12()117(6.36

×=r

13689117=r

41301201=r

56.14836.36=r

117117=r

2.203201=r

52.386.36=r

r = 1 r = .99 r = .95


4. Computation of Covariance

Once the correlation coefficient is determined as above, it is possible to compute the covariance between the market return and the return on each of stocks 1, 2, and 3 pursuant to the following formula:

Covariance Between Stock 1 and the Market = Cov (1m) = σ1m = COR1mσ1σm

This formula says that the covariance between stock 1 and the market can be computed by multiplying the correlation coefficient between stock 1 and the market by the standard deviation of stock 1 and the standard deviation of the market as follows:

Table BB

Covariance Between the Three Stocks and the Market [1] [2] x [3] x [4] = [5]

Stock Correlation Coefficient

Between Stock in [1] and the Market

COR1m = r1m

x

Standard Deviation of Stock in [1]

σ1,2 or 3

x

Standard Deviation of

Market

σm

=

Covariance Between

Stock in [1] and the Market

Stock 1

1

x

6.68

x

6.78

=

46

Stock 2

.98

x

11.57

x

6.78

=

76.87

Stock 3

.95

x

2.24

x

6.78

=

14.42

Now it is possible to compute the beta.

E. The Determination of Beta

The beta of stock 1 is derived by dividing (1) the covariance between (a) the returns on stock 1, with (b) the returns on the market portfolio (COR1mσ1σm), by (2) the variance of the market portfolio (σm

2). The formula can be written as follows:96

2

11mCOR 1Stock of m

m

σσσβ =

This formula says that the beta of stock 1 is equal to (1) the product of (a) the correlation coefficient between stock 1 and the market (COR1m), (b) the standard deviation of returns

96. See BREALEY & MYERS, supra note 1, at 145.


on stock 1 (σ1), and (c) the standard deviation of returns on the market (σm), divided by (2) the variance of the market (σm

2). Thus, the beta of the market portfolio is equal to one. This can be established as

follows:

2

11mCOR m

m

σσσ

2

1 m

mm

σσσ=

2

2

m

m

σσ= = 1

Comparatively, the beta of a risk-free asset, such as a T-bill, is equal to zero because there is no correlation between the price of treasury bills and the price of the market. This can be established as follows (T represents treasury bills):

2

TmCOR m

mT

σσσ

2

0m

mT

σσσ= = 0

The beta of a particular stock, which can be negative, zero, or positive, is an indicator of the degree to which the price of the stock changes in relation to changes in the market.

The betas for stocks 1, 2, and 3 can be determined by dividing the covariances of these stocks as determined above by the variance of the market portfolio as determined above:

2

),(m

yxCovσ

β =

Table CC Computation of Beta of the Three Stocks

[1] [2] [3] [4]

Stock Covariance of

Stock in [1] with Market

(See Table BB)

÷

Variance of Market

σm2

(See Table Q)

=

Beta of Stock in [1]

Stock 1 46 ÷ 46 = 1 Stock 2 76.87 ÷ 46 = 1.67 Stock 3 14.42 ÷ 46 = .31

This shows that if the market goes up or down by 10%, the price of stock 1 is expected to go up or down by 10%, the price of stock 2 is expected to go up or down by 16.67%, and the price of stock 3 is expected to go up or down by 3.1%. Beta is a much more sensitive measure of the relationship between the returns on stocks and the market than the correlation coefficient. Although the correlation coefficients for stocks 1, 2, and 3 are very close at 1, 0.98, and 0.95, respectively, the betas for these stocks are dramatically different at 1, 1.66, and 0.31, respectively.


IV. DERIVATION OF BETA THROUGH REGRESSION ANALYSIS

A. Introduction

Beta can also be determined through the use of simple regression analysis. Regression analysis uses past data on the relevant independent and dependent variables to develop a predictive equation.97 The independent or explanatory variable, in this case the market return, is used to make a prediction about the value of the dependent variable, in this case the return on stocks 1, 2, and 3.

Like the correlation coefficient, regression analysis provides an estimate of the degree of correlation between the dependent and independent variables, thereby measuring the strength of the relationship.98 With simple regression analysis, which is based on one independent variable, the equation for predicting the dependent variable (y) is a linear function of a given independent variable (x). The prediction equation is the familiar equation for a straight line:

bxay +=ˆ

The constant a is the intercept which is the predicted value of y (e.g., return on stock 1) if the predicted value of x (i.e., return on the market) is zero.99 Simple linear regression uses past data to fit a prediction line that relates a dependent variable (y) (e.g., the returns in stocks 1, 2, and 3) to a single independent variable (i.e., the return on the market).100 A fundamental assumption of simple regression analysis is that “the relation is, in fact, linear. According to the linearity assumption, the slope of the equation [i.e., b] does not change as x changes.”101

B. Scatter Diagrams

One way of making a judgement about whether a linear relationship exists between the dependent variable Y and the independent variable X is to construct a scatter diagram that shows the location of the points (X,Y) on a graph in which the horizontal axis represents the independent variable X and the vertical axis indicates the dependent variable Y. Murray R. Spiegel explains that “If all points in [the] scatter diagram seem to lie near a line, . . . the correlation is called linear. In such cases, . . . a linear equation is appropriate for purposes of regression (or estimation).”102 Scatter diagrams for stocks 1, 2, and 3 and the market are set out in the following graphs. These diagrams show the relationship between the market’s return and the return for the particular stock.

97. HILDEBRAND & OTT, supra note 1, at 419; see also GUJARATI , supra note 1, at chs. 1-4. 98. HILDEBRAND & OTT, supra note 1, at 419. 99. Id. at 420. 100. Id. at 421. 101. Id. 102. SPIEGEL, supra note 1, at 294.


GRAPH C

Scatter Diagram for Market Return and Return on Stock 1

(See Table W)

Relationship Market Stock 1

15% 15% 10% 10% 0% 0%

Obviously, the relationship is linear.

GRAPH D


(See Table W)

Relationship

Market Stock 2 15% 20% 10% 15% 0% -5%

The relationship is linear, but less so than for stock 1.


GRAPH E


(See Table V)

Relationship Market Stock 3

15% 10% 10% 7% 0% 5%

The relationship is also linear. Thus, it would appear that regression analysis would be appropriate for relating the return on the market to the returns on each of stocks 1, 2, and 3.

C. The Method of Least-Squares

The least-squares method is used to determine a “best fitting line” that relates the independent and dependent variables . As explained by Hildebrand and Ott:

The regression analysis problem is to find the best straight-line prediction. The most common criterion for ‘best’ is based on squared prediction error [i.e., the vertical deviations from the line] . . . . [T]he equation for the prediction line [is] the slope [b] and intercept [a] that minimize the total squared prediction error.103

D. Normal Equations for Least-Squares Line

As indicated, the equation for the least-squares line can be set out as follows:

y = a + bx

The constants, the intercept (a) and the slope (b), are determined by solving simultaneously the equations: 103. HILDEBRAND & OTT, supra note 1, at 429-30.


Σy = aN + bΣx Σxy = aΣx + bΣx2

These are called the normal equations for the least-squares line.104 The values of the various terms of these equations can be determined by the use of a

table as set out below for stock 1 in Table DD.105

Table DD Least-Squares Table

for Stock 1 [1] [2] [3] [4] x

Market’s Return y

Stock’s Return x2

[1] squared xy

[1] x [2] 15% 15% 225 225 10% 10% 100 100 0% 0% 0 0

Σx = 25% Σy = 25% Σx2 = 325 Σxy = 325 Because there are just three pairs of values of x and y, N=3, the least-squares equation becomes:

Least-Squares Equation

Σy = aN + bΣx (25) = a(3) + b(25) 25 = 3a + 25b

and

Σxy = aΣx + bΣx2 (325) = a(25) + b(325) 325 = 25a + 325b

Next, the two equations must be solved simultaneously. First, the slope b is eliminated so that the intercept can be computed. This is accomplished by multiplying the first equation by -13 and then subtracting the second equation, thereby eliminating the slope.

104. SPIEGEL, supra note 1, at 297-98; see also GUJARATI , supra note 1, at 55. 105. SPIEGEL, supra note 1, at 275.


Solving the Equation Simulataneously

(-13)25 = (-13)3a + (-13)25b

= -325 = -39a - 325b [minus]

325 = 25a + 325b

= 0 = -14a = 0

a = 0/14

a = 0 Now that the intercept a is found, the slope can be found as follows:

Finding the Slope

25 = 3a + 25b 25 = 3(o) + 25b 25 = 25b b = 25/25 b = 1

Thus, the equation for the least-squares line can be determined as follows:

Regression Line Equation for Stock 1

y = a + bx y = 0 + (1)x y = x

This b = 1 is the beta for stock 1, and it can be graphed as follows, with x = 5%, 10%, 15%, and 20%:


GRAPH F

Beta for Stock 1

Value of x Regression

Line Equation

y = x x = 5% 5% = 5%

x = 10% 10% = 10% x = 15% 15% = 15% x = 20% 20% = 20%

The least-squares tables for stocks 2 and 3 are set out below:


Table EE Least-Squares Table for Stock 2

[1] [2] [3] [4]

X Market's Return

Y Stock's Return

X2 [1] squared

XY [1] x [2]

15% 20% 225 300

10% 15% 100 150

0% -5% 0 0

25% =x ∑ 30% =y ∑ 325 = x2∑ 450 =xy ∑

Table FF Least-Squares Table for Stock 3

[1] [2] [3] [4]

X Market's Return

Y Stock's Return

X2 [1] squared

XY [1] x [2]

15% 10% 225 150

10% 7% 100 70

0% 5% 0 0

25% =x ∑ 22% =y ∑ 325 = x 2∑ 220 =xy ∑

The least squares equations for s tocks 2 and 3 are as follows:

Least-Squares Equations

Stock 2 Stock 3

1.

25b + 3a = 30 b(25) + a(3) = 30

xb + aN =y ∑∑

25b + 3a = 22 b(25) + a(3) = 22

xb + aN =y ∑∑

and

2.

325b + 25a = 450 b(325) + a(25) = 450

xb +x a =xy 2∑∑∑

325b + 25a = 220 b(325) + a(25) = 220

xb +x a =xy 2∑∑∑


Next, the two equations must be solved simultaneously by first eliminating the slope so that the intercept can be computed. For both equations, this is done by multiplying the first equation by -13 so that when the second equation is subtracted from the first, slope (b) is eliminated.

Solving the Equation Simultaneously

Stock 2 Stock 3

Determining the Intercept Determining the Intercept (-13)30 = (-13)(3a) + (-13)25b (-13)22 = (-13)(3a) + (-13)25b

= -390 = -39a - 325b = -286 = -39a -325b [minus] [minus]

450 = 25a + 325b 220 = 25a + 325b

= 60 = -14a + ob = -66 = -14a + ob a = 60 -14

a = -66 -14

a = -4.28 a = 4.71

Now that the intercepts for stocks 2 and 3 have been determined, the slopes (b) for each of these stocks can be obtained as follows:

Finding the Slopes (the Beta)

Stock 2 Stock 3 Determining the Slope Determining the Slope

30 = (3a) + 25b 22 = 3a + 25b 30 = 3(-4.28) + 25b 22 = 3(4.71) + 25b 30 = -12.84 + 25b 22 = 14.13 + 25b

42.84 = 25b 7.87 = 25b b = 42.84

25 b = 7.87

25 b = 1.71 b = .314

It should be noted that this slope is the beta for stocks 2 and 3. The beta for stock 2 is slightly over the 1.66 beta for stock 2, determined statistically through the use of covariance, and the beta for stock 3 is the same as the .31 beta for stock 3, determined statistically through the use of covariance.

The equations for the regression lines for stocks 2 and 3 are as set forth below:

Regression Line Equations

Equation for Stock 2 Equation for Stock 3 y = -4.28 + 1.7x y = 4.71 + .31x

These two equations can be graphed as follows with, in both cases, x = 0%, 10%, and 15%. The actual returns are also shown.


GRAPH G Best Fit Line for Stock 2

Value of x Regression Line Equation

y = -4.28 + 1.7x x = 0 y = -4.28 + 1.7(0) y = -4.28% x = 10% y = -4.28 + 1.7(10) y = -4.28 + 17.0 y = 12.72% x = 15% y = -4.28 + 1.7(15) y = -4.28 + 25.5 y = 21.22%


GRAPH H Best Fit Line for Stock 3

Value of x Regression Line Equation

y = 4.71 + .31x x = 0 y = 4.71 + .31(0) y = 4.71 x = 10% y = 4.71 + .31(10) y = 4.71 + 3.1 y = 17.81 x = 15% y = 4.71 + .31(15) y = 4.71 + 4.65 y = 9.36

E. Short-Cut Method For Computing Slope-Beta

A short-cut way of computing the slope or beta is to use the following formula:106

Short-Cut Method

SxxSxy

= x

xy = b

2∑∑

106. HILDEBRAND & OTT, supra note 1, at 430; see also GUJARATI , supra note 1, at 55.


In this formula, x equals x minus the expected value of x(x - x ) and y equals y minus the expected value of y(y - y ).

The following short-cut beta tables are set up to compute Σxy and Σx2:

Table GG Short-Cut Beta Table for Stock 1

[1] [2] [3] [4] [5] [6] x

Market Return

y Stock Return

x - x x = 8%

y - y y = 8%

(x - x )(y - y ) [3] x [4]

(x - x )2 [3]squared

15% 15% 7% 7% 49 49 10% 10% 2% 2% 4 4 0% 0% -8% -8% 64 64

Σ(x - x )(y - y ) = 117 Σ(x - x )2 = 117

Table HH Short-Cut Beta Table for Stock 2

[1] [2] [3] [4] [5] [6] x

Market Return

y Stock Return

x - x = 8%

y - y = 9%

(x - x )(y - y ) [3] x [4]


15% 20% 7% 11% 77 49 10% 15% 2% 6% 12 4 0% -5% -8% -14% 112 64

Σ(x - x )(y - y ) = 201 Σ(x - x )2 = 117

Table II Short-Cut Beta Table for Stock 3

[1] [2] [3] [4] [5] [6] x

Market Return

y Stock Return

x - x = 8%

y = y = 7.4%

(x - x )(y - y ) [3] x [4]


15% 10% 7% 2.6% 18.2 49 10% 7% 2% -.4% .6 4 0% 5% -8% -2.4% 19.2 64

Σ(x - x )(y - y ) = 201 Σ(x - x )2 = 117

Using the simplified formula for computing beta and the information from the above

beta tables, the results are as follows:


Computing Betas with the Shortcut Method

Beta Stock 1 Beta Stock 2 Beta Stock 3

2b

x

xy

ΣΣ=

2b

x

xy

ΣΣ=

2b

x

xy

ΣΣ=

117117b =

117201b =

1172.38b =

b = 1 b = 1.71 b = .32

These are close to the betas obtained from the above calculation of the least-squares line. Once the slope is determined using the short-cut formula above, the intercept can be computed with the use of the following short-cut formula:

a = y - b x For the three stocks the intercepts are computed as follows:

Computing Intercepts of the Three Stocks

Stock 1 Stock 2 Stock 3 a = y - b x a = y - b x a = y - b x a = 8 - 1(8) a = 9 - 1.71(8) a = 7.4 - .31(8)

a = 0 a = 9 - 13.68 a = 7.4 - 2.48 a = -4.68 a = 4.92

Using the normal equations for the least squares, the intercepts are 0, -4.28, and 4.71, respectively. The intercept shows the expected return for the stock if the market's return is zero.

F. Coefficient of Determination

The coefficient of determination (r2) is a measure of the strength of the correlation between a dependent variable (Y) and an independent variable (X). It “measures the proportion or percentage of the total variation in [the independent variable] Y explained by the regression model.”107 The coefficient of determination is derived from an analysis of (1) the variance of the actual values of Y from the expected value of the dependent variable (Y ), and (2) the variance of the actual values of Y from the estimate (determined from the equation of the regression line of Y on X) of the values of the dependent variable (Yest).108 Specifically, the total variation of Y is defined as:

Total Variation = Σ(Y - Y )2

107. GUJARATI , supra note 1, at 77. 108. SPIEGEL, supra note 1, at 296.


This equation says that the total variation is the “sum of the squares of the deviations of the values of Y from the mean Y .”109

The total variation is made up of the explained variation and the unexplained variation. The explained variation is defined as the sum of the squares of the deviations of (1) the estimated values of Y (Yest) determined from the regression line, from (2) the mean of Y ( Y ). Thus, the explained variation can be written as follows:

Explained Variation = Σ(Yest - Y )2

On the other hand, the unexplained variation is defined as the sum of the squares of the deviations of (1) the actual values of Y, from (2) the estimated values of Y (Yest) determined from the regression line. Thus, the unexplained variation can be written as follows:

Unexplained Variation = Σ(Y - Yest)2

The deviations of Y from Yest are referred to as unexplained variations because these deviations “behave in a random or unpredictable manner.”110

To summarize, total variation, is the sum of the explained variation and the unexplained variations, and is written as follows:

Total Variation = Explained Variation + Unexplained Variation

Σ(Y - Y )2

=

Σ(Yest - Y )

+

(Y - Yest)

The coefficient of determination (r2) is the ratio of explained variation to total variation, that is:111

2

2

) -(Variation Total)-(Variation Explained

YYYYest

rΣΣ=

The ratio lies between 0 and 1. If r2 is zero, this means there is no explained variation and the total variation is, therefore, all unexplained—that is, Y is not dependent at all on X.112 Thus, an r2 of 1 means there is perfect correlation between the dependent and independent variables . This is because each estimate of the value of Y (Yest), from the regression line, is the same as the actual value of Y. Tables JJ, KK, and LL below compute the r2 for each of stocks 1, 2, and 3.

109. Id. 110. Id. 111. Id. 112. Id.


Table JJ (r2) Coefficient of Determination for Stock 1

[1]

X

[2]

Y

[3]

See Table F

[4]

Yest (Estimate from

Regression Line with X=0, 10%, 15%) (See Graph F)

[5]

Total Variance (Y - Y )2

(See Table N)

[6]

Explained Variance (Yest - Y )2 ([4] - [3])2

15% 15 8 15 49 (15 - 8)2 = (7)2 = 49 10% 10 8 10 4 (10 - 8)2 = (2)2 = 4 0% 0 8 0 64 (0 - 8)2 = (-8)2 = 64

Σ(Y - Y )2 = 117 Σ (Yest - Y )2 = 117

117 ) -(Variation Total117 )-(Variation Explained

2

22

=Σ=Σ=

YYYYest

r

1171172 =r

r 2 = 1

Table KK

(r2) Coefficient of Determination for Stock 2 [1]

X

[2]

Y

[3]

See

Table G

[4]

Yest (Estimate from

Regression Line with X=0, 10%, 15%) (See Graph G)

[5]


(See Table O)

[6]


15% 20 9 21.22 121 (21.22 - 9)2 = (12.22)2 = 149.33

10% 15 9 12.72 36 (12.72 - 9)2 = (3.72)2 = 13.84 0% -5 9 -4.28 196 (-4.28 - 9)2 = (-13.28)2 =

176.36 Σ(Y - Y )2 = 353 Σ(Yest - Y )2 = 339.53

533 ) -(Variation Total39.533 )-(Variation Explained

2

22

=Σ=Σ=

YYYYest

r

35353.3392 =r

r 2 = .96


TABLE LL

(r2) Coefficient of Determination for Stock 3 [1]

X

[2]

Y

[3]

See Table

H

[4]

Yest (Estimate from

Regression Line with X=0, 10%,

15%) (See Graph H)

[5]


(See Table P)

[6]


15% 10 7.4 9.36 6.76 (9.36 - 7.4)2 = (1.96)2 = 3.84 10% 7 7.4 7.81 0.16 (7.81 - 7.4)2 = (0.41)2 = 0.17 0% 5 7.4 4.21 5.76 (4.71 - 7.4)2 = (-2.69)2 = 7.24

Σ(Y - Y )2 = 12.68 Σ(Yest - Y )2 = 11.25

12.68 ) -(Variation Total1.251 )-(Variation Explained

2

22

=Σ=Σ=

YYYYest

r

68.1225.11r 2 =

r 2 = .89

As demonstrated in Table JJ, the r2 for stock 1 is 1, meaning that all of the variance

for stock 1 is explained by the movements in the market. Table KK shows that the r2 for stock 2 is 0.96, meaning that only 4% of the variance for stock 2 is not explained by movements in the market, and Table LL shows that the r2 for stock 3 is 0.89, meaning that 11% of the variance for stock 3 is not explained by movements in the market.

The coefficient of correlation (or correlation coefficient (r)) is simply the square root of r2, as follows:113

2

2

) -(Variation Total)-(Variation Explained

YYYYest

rΣΣ±=

The values of r vary from -1 and +1, with the minus sign indicating a negative linear correlation and the positive sign indicating a positive linear correlation.114 It should be noted that “r is a dimensionless quantity; that is, it does not depend on the units employed.”115

113. SPIEGEL, supra note 1, at 297. 114. Id. 115. Id.


If r has a value of zero, there is no linear correlation between the two variables. On the other hand, if r is near 1 or -1 the correlation is said to be high.116 However, even if the correlation is high, there may be no causation, because the correlations may be nonsense or spurious.117 The correlation coefficient “measures the goodness of fit between (1) the equation actually assumed and (2) the data.”118

As indicated in Part III.D.3 above, the correlation coefficient (r) can also be written as follows:

))-()()-(()()(

22

22

yyxx

yyxxr

ΣΣ

−−Σ=

))(( 22 yx

xyr

ΣΣ

Σ=

The computation of the correlation coefficients for the three stocks pursuant to this formula are set out in Table AA. There we see that the r’s are as follows:

Stock r

1 1.00 2 0.98 3 0.95

In computing r2 from these r’s one gets:

Stock r2 = r2

1 12 = 1 2 0.982 = 0.96 3 0.952 = 0.90

And the r2s computed from the two formulas are as follows:

Stocks r2 Computed from Formula for

Coefficient of Determination r2 Computed by Squaring the r Determined

from the Formula in Section III.D.3. 1 1.00 1.00 2 0.96 0.96 3 0.89 0.90

The difference in the computation of the r2 for stock 3 is presumably attributable to rounding errors.

116. Id. 117. Id. 118. SPIEGEL, supra note 1, at 297.


V. DERIVATION OF BETA THROUGH THE USE OF EXCEL’S LINEAR REGRESSION FUNCTION

Beta can be computed simply through the use of the linear regression function on a computer program, such as the LINEST statistical function in the Microsoft Excel program. The first step in using the function is to specify the relationship between the returns on the market and the returns on three stocks . These relationships are set out below in Tables MM through OO:

Table MM

Excel Linear Regression Stock 1

Table NN


Table OO


X Market Return

Y Stock 1 Return

X Market Return

Y Stock 2 Return

X Market Return

Y Stock 3 Return

15% 15% 15% 20% 15% 10% 10% 10% 10% 15% 10% 7% 0% 0% 0% -5% 0% 5%

Once the data is arrayed as above, beta for each stock can be determined by the insertion of the following formula in the formula bar:

= INDEX(LINEST (known ys, known xs),1)

Thus, for example, assume that in Table MM the returns for stock 1, the y or dependent variable, are set out in column and rows C6 through C8 on an Excel worksheet, and the returns for the market, the x or independent variable, are set out in column and rows B6 through B8. Beta could then be computed in, for example, cell B10 by putting the following formula in the formula bar for cell B10:

=INDEX(LINEST(C6:C8, B6:B8),1)

As a result the beta of 1 appears in cell B10. The intercept can be determined using the following formula:

=INDEX(LINEST(known ys, known xs),2)

The betas using the LINEST function are as follows:


Table PP Betas Using LINEST Function Stock Beta

1 1.00 2 1.71 3 0.31

VI. SUMMARY OF BETA RESULTS AND INTRODUCTION TO ADJUSTED BETAS

A. Summary

In summary, the betas determined under the above four methods ((1) use of covariance and variance, (2) use of the normal equations for the least-squares line, (3) use of the shortcut regression method, and (4) use of Excel linear regression formula), result in the following betas:

Table QQ

Summary of Computations of Beta Estimated Beta

for Stocks

Method 1 2 3

[1] Covariance Divided by Variance

2

),(âm

yxCOVσ

=

1

1.67

.31

[2] Normal Equations for the Least-Squares Line Σx = aN + bΣx

Σxy = aΣx + bΣx2

1

1.71

.31

[3] Short-Cut Formula

2bxxy

ΣΣ=

1

1.71

.32

[4] Excel: Linear Regression Function

=INDEX(LINEST(known ys, known xs),1)

1

1.71

.31

The largest difference is in the computation of beta for stock 2. Using the covariance

divided by variance method provides a somewhat different result than any of the other methods. The difference is presumably attributable to rounding differences . If the betas are rounded to one decimal point, each method produces a beta of 1.7 for stock 2.


B. Adjusted Betas

True betas, as measured above, tend to move toward 1 over time .119 This has led to the development of adjusted betas. Brigham and Gapenski explain, “One can begin with a firm’s pure historical statistical beta, make an adjustment for the expected future movement toward 1.0, and produce an adjusted beta which will, on average, be a better predictor of the future beta than would the unadjusted historical beta.”120 They go on to report that ValueLine’s adjusted betas are based on approximately this formula:

Adjusted Beta = 0.33 (Historical Beta) + 0.67(1.0).

This concept is not examined further here.

VII. ASSET AND EQUITY BETAS AND THE LEVERING AND UNLEVERING FORMULA

A. Introduction121

A firm’s cost of capital is a function of its business risk—the risk associated with the firm’s investments or assets.122 In a firm with no debt, the common shareholders face only this business risk. But if the firm carries debt (i.e., is leveraged), the common shareholders also face financial risk: the risk that the firm will not be able to service the debt and therefore will go into bankruptcy.123 Obviously, the more leverage, the greater the risk to the common shareholders. Financial risk, however, does not affect business risk; leverage does not affect the firm’s assets.124

In determining the beta of a firm’s equity, the beta must be adjusted to reflect the financial risk (i.e., leverage) employed by the particular company. A firm’s leverage controls its level of financial risk and its beta. As a firm’s leverage increases, its financial risk and beta correspondingly increase.125 A firm’s asset beta would apply if the firm did not employ any leverage. Therefore, the asset beta assumes a firm incurs business risk, but not financial risk.

Brealey and Myers explain that because low risk is generally associated with debt of large publicly-held firms with little debt, financial analysts generally assume that the beta of the debt of such firms is zero . When the firm’s debt beta is zero, its debt is not sensitive to moves in the equity market.126 This concept of stability is intuitively appealing. Debt holders in firms that are not highly leveraged can be fairly certain that they will be paid regardless of market shifts . Assuming Brealey and Myers are correct, the value of the debt is not affected by market swings. Brealey and Myers explain, however, that stability may not occur in periods of volatile interest rates such as the early

119. BRIGHAM & GAPENSKI, supra note 1, at 353. 120. Id. 121. This Part is excerpted from Thompson, supra note 1, at 514-17. 122. Id. at 514 (citing BREALEY & MYERS, supra note 19, at 189). 123. Thompson, supra note 1, at 514 (citing BREALEY & MYERS, supra note 19, at 189). 124. Thompson, supra note 1, at 514 (citing BREALEY & MYERS, supra note 19, at 190). 125. Thompson, supra note 1, at 514 (citing BREALEY & MYERS, supra note 19, at 191-92 and THOMAS E. COPELAND & J. FRED WESTON, FINANCIAL T HEORY AND CORPORATE POLICY 455-60 (3d ed. 1988)). 126. Thompson, supra note 1, at 514 (citing BREALEY & MYERS, supra note 19, at 191).


1980s when betas on corporate bonds were as high as 0.3 to 0.4.127 In a highly leveraged firm, it can be expected that the value of the debt will change with changes in the market. However, regardless of whether the value of a firm’s debt fluctuates or remains constant, investing in the firm’s debt will be less risky than holding equity in the same firm.

If the beta of a firm’s debt is zero, the beta of its stock will equal the firm’s asset beta. The asset beta is equal to the sum of the weighted average of the beta of the firm’s debt (D), and the beta of the firm’s equity (E).128 Thus, the firm’s asset beta can be expressed algebraically as:

equity of VEdebt of

VDassets of β+β=β

In this formula, VD is the portion of the firm’s value attributable to debt, and

VE is the

portion of the value attributable to equity. If the beta of the firm’s debt is positive (meaning the debt is sensitive to market

moves), the beta for the firm’s equity will exceed the firm’s asset beta because of the financial risk associated with the debt. The weighted averages of the firm’s debt beta and equity beta will equal the firm’s asset beta (i.e., the beta of a nonleveraged or low leveraged firm).

The following illustration helps conceptualize these ideas.129 Assume that a nonleveraged firm has an asset beta of 0.8. This is also the firm’s equity beta as illustrated on the following graph:

127. Thompson, supra note 1, at 514 (citing BREALEY & MYERS, supra note 19, at 191 n. 9). 128. Thompson, supra note 1, at 514 (citing BREALEY & MYERS, supra note 19, at 191). 129. Thompson, supra note 1, at 515 (citing BREALEY & MYERS, supra note 19, at 191-92 (providing the basis for the illustration)).


GRAPH I Asset Beta

This means that without any debt the stock is not very sensitive to market moves—the price of the equity will move both up and down less than the market.

Now assume that the firm takes on such substantial debt that debt represents 40% of the value of the firm and the debt has a beta of 0.2. The firm’s equity beta, which is now 1.2, can then be computed algebraically since the only unknown in the following formula is the equity beta:

)equity( of âVEdebt of â

VDassets â x+=

0.8 = (0.4)(0.2) + (0.6)(x)

0.8 = 0.08 + (0.6)(x)

x=6.08.-8.

x=6.72.

1.2 = x


This result is intuitively appealing because it can be expected that, as a firm becomes more and more leveraged, the equity holders will demand a higher return to compensate for the added financial risk.

The relationship between the debt and asset betas in the above case can be diagramed as:

GRAPH J Relationship Between Asset Beta and Equity Beta

The following principles can be derived by relating the previous diagram to the following formula for determining the asset beta:

equity of VEdebt of

VDassets βββ +=

First, the lower the beta of debt, the lower the beta of equity. If debt has a beta of zero, the beta of equity is equal to the asset beta. Second, the higher the beta for debt, the higher the beta for equity. Third, the asset beta equals the weighted average of the firm’s debt and equity betas. Fourth, the firm’s asset beta does not change with leverage, as asset betas are not affected by financial risk. This, too, is intuitively appealing because the income stream does not depend on whether the cash flow pays the firm’s debt holders or equity holders.

Many industry-wide betas reflect asset betas after removing the effects of financial leverage on beta. Thus, determining the equity beta of a particular firm in the industry


requires adjusting the asset beta upward for any leverage utilized.130 However, if an equity beta is available, it may be necessary to convert the equity beta into an asset beta, a process referred to as “unlevering.”131 The levering and unlevering formula is discussed in next.

B. The Levering and Unlevering Formula132

If a firm has no debt (i.e., leverage) and only one type of asset (e.g., railroad operations) the firm’s equity beta will also be its asset beta, and the equity beta could be expressed as the unlevered equity beta, βu. If, however, the firm has debt, the firm’s equity beta will reflect this leverage and can be expressed as a leveraged equity beta, β l.

In a 1969 article, Robert Hamada set out a mathematical formula for moving between levered and unlevered betas,133 and the formula is commonly used today.134 As explained by Brigham and Gapenski, “Robert Hamada combined the Capital Asset Pricing Model . . . with the MM [i.e., Modigliani-Miller] after-tax [capital structure] model to obtain” the cost of equity capital for the levered firm (rle).135 Hamada’s formula is as follows:136

Cost of Levered Equity

= Risk-free Rate

+ Business Risk Premium

+ Financial Risk Premium

rle = rf + βu(rm - rf) + βu(rm - rf)(1-T)(D/E)

T is the combined federal and state corporate income tax rate (assume 40%), and D/E is the firm’s debt to equity ratio. Thus, if the firm has no debt, the financial risk term becomes zero and the firm’s cost of equity, rle, is determined by the standard form CAPM, with beta being the unlevered beta, βu.

Hamada’s formula divides the required cost of equity capital for the levered firm (rle) into three elements:137 (1) a risk-free rate, rf; (2) a premium for business risk, βu(rm - rf); and (3) a premium for financial risk, βu(rm - rf)(1-T)(D/E).138 Hamada also derived from the above formula a formula for determining the beta for a levered firm, β l, as follows. The starting point is the basic formula for the cost of equity for the levered firm (rle):

rle = rf + âu(rm - rf) + βu(rm - rf)(1-T)(D/E)

130. Thompson, supra note 1, at 517 (referring to BREALEY & MYERS, supra note 19, at 190-92). 131. Thompson, supra note 1, at 517 (citing SHAPIRO , supra note 50, at 263-65, which discussed the unlevering process). 132. See BRIGHAM & GAPENSKI, supra note 1, at 488-91, 629-31. 133. Robert S. Hamada, Portfolio Analysis, Market Equilibrium, and Corporate Finance, 24 J. FIN. 13 (1969). 134. See BRIGHAM & GAPENSKI, supra note 1, at 488-91, 629-31. 135. Id. at 629. 136. Id. 137. Id. 138. Id.


This formula can be simplified as follows to arrive at levering and delevering formulas:

First: Set the required levered equity return into a form that is dependent on a levered beta, β l:

rle = rf + β l(rm - rf)

Second: Set the right-hand term equal to the Hamada form of the cost of levered equity reflecting separate premiums for business and financial risk:

rf + β l(rm - rf) = rf + βu(rm - rf) + βu(rm - rf)(1-T)(D/E)

Third: Eliminate the rf term from both sides of the equation:

β l(rm - rf) = βu(rm - rf) + βu(rm - rf)(1-T)(D/E)

Fourth: Divide both sides of the equation by (rm - rf):

β l = βu + βu(1-T)(D/E)

Fifth: Simplify the right-hand side of the equation:

β l = βu[1 + (1-T)(D/E)]

Thus, the levered equity beta for a firm (β l) can be determined from the unlevered equity beta for the firm (βu), pursuant to the following formula:

Levering Formula

β l = βu[1 + (1-T)(D/E)]

and, correlatively, the unlevered equity beta for the firm (βu) can be derived from the levered equity beta (β l) pursuant to the following formula:

Unlevering Formula

T)(D/E)]-(1 [11

+= ββu

Brigham and Gapenski sum up these principles as follows: Thus, under the MM and CAPM assumptions, the beta for any firm is equal to the beta the firm would have if it used zero debt, adjusted upward by a factor that depends on (1) the corporate tax rate and (2) the amount of financial leverage employed. Therefore, the firm’s market risk, which is measured by [β l],


depends on both the firm’s business risk as measured by βu and its financial risk as measured by [β l] - βu = βu(1-T)(D/E).139 In many cases it will be possible to determine a firm’s equity levered beta (β l)

before determining a firm’s asset beta. The equity levered betas could then be delevered to get the unlevered equity beta or asset beta pursuant to the following formula:

T)(D/E)]-(1 [11

+= ββu

In other cases, the analyst may have an estimate of a firm’s asset beta (βu) and will have to lever up that beta to determine the firm’s equity levered beta (β l), pursuant to the following formula:

β l = βu[1 + (1-T)(D/E)]

One way of getting an estimate of the beta for an asset (e.g., a railroad operation) is to use what Brigham and Gapenski call the “pure-play method.”140 Under this method, several publicly traded firms that are only engaged in the railroad business are identified and the average leverage equity beta for these firms is determined by regressing their stock returns against the market. This estimate of the leverage equity beta is then delevered using the average debt to equity ratio of those firms in the delevering formula:

T)(D/E)]-(1 [11

+= ββu

As will be seen in Part IX, this pure play method was used by Lazard in determining the cost of capital for Conrail. Lazard first estimated Conrail’s levered equity beta (β l) by, for example, regressing the historical returns on the stock of Conrail and several other railroad firms against the historical returns on the market.141 Lazard then delevered the beta using the above delevering formula,142 and then relevered the beta for various capital structure assumptions using the following levering formula:143

β l = βu[1 + (1-T)(D/E)]

VIII. RETURNING TO THE CAPITAL ASSET PRICING MODEL144

Returning to the CAPM model, the elements of CAPM are (1) the beta for the particular stock; (2) the risk-free rate (rf); and (3) the market risk premium (rm-rf). A 139. BRIGHAM & GAPENSKI, supra note 1, at 630. 140. Id. at 490-91. 141. See infra Appendix A, Columns [1], [2], and [3]. 142. See infra Appendix A, Column [4]. 143. See infra Appendix A, Column [9]. 144. This Part is excerpted from Thompson, supra note 1, at 511-13.


market portfolio of common stocks has a beta of 1 because the “covariance of the market itself is identical to the variance of the market portfolio.”145 The expected risk premium of holding a market portfolio is equal to the market risk premium (rm-rf), and the expected return of holding the market portfolio is equal to the sum of the risk-free rate (rf) and the market risk premium (rm– rf). Thus, there is a one-to-one relationship between the beta of a portfolio of common stocks and the expected return of such a portfolio. The following graph illustrates this one-to-one relationship:

GRAPH K

Beta of Portfolio of Common Stocks

This graph shows that the beta of a risk-free Treasury bill, which has an expected

return of rf, is zero, and the beta for the market portfolio, which has a return of rm, is 1. The graph also illustrates that any security with a beta of less than 1 can expect a return that is less than the market return because such a security is less risky than the market portfolio. Furthermore, any security with a beta greater than 1 has an expected return that is more volatile than that of the market because that security is more risky than the market portfolio. Thus, the cost of capital for a low beta stock will be less than the cost of capital for the market, and the cost of capital of a high beta stock will be higher than the cost of capital for the market. For example, AT&T, which has a beta of less than 1, will have a lower cost of capital than Tandem Computer, which has a beta greater than 1.

The line intersecting the risk-free rate on the vertical axis is known as the security market line (SML). The SML illustrates the capital asset pricing model, which provides that “the expected risk premium on each investment is proportional to its beta.”146

145. Id. at 511 (citing COPELAND & WESTON, supra note 125, at 198). 146. Thompson, supra note 1, at 512 (quoting BREALEY & MYERS, supra note 19, at 162).


GRAPH L Security Market Line

The expected risk premium on a particular stock can, therefore, be written as:147

Expected Risk Premium = (β) x (Expected Risk Premium Market) = β x (rm - rf) = β (rm - rf)

The expected rate of return (or the discount rate, hurdle rate, or cost of capital) can be derived from the algebraic statement of the CAPM:

Expected Rate of Return = r = (Risk-Free Rate of Return) + (β) x Market Risk Premium r = rf + β (rm - rf)

In a well functioning capital market, “a security cannot sell for an extended period at prices low enough to yield more than the appropriate return indicated on the SML.”148 Such a security would become an attractive investment relative to other securities with similar risk. The demand for the security would cause its price to rise until the expected return fell to the appropriate point on the SML. Conversely, a stock with a price high enough to put its expected return below the appropriate point on the SML would induce investors to sell the stock, thereby driving down the price until the rate of return was appropriately placed on the SML.149 147. Thompson, supra note 1, at 512 (citing BREALEY & MYERS, supra note 19, at 138). 148. Thompson, supra note 1, at 513 (quoting Mullins, supra note 52, at 108). 149. Thompson, supra note 1, at 513 (citing Mullins, supra note 52, at 108).


Thus, in using CAPM to determine the appropriate discount rate for the acquisition of a target corporation, it is necessary to identify: (1) the risk-free rate (i.e., the rate on Treasury bills), (2) the market risk premium, which over time has been 8.4%, and (3) the beta for the target corporation. Although there is an element of judgement in determining beta, most estimates should be in the same ballpark.

If the target is involved in several lines of business, it is appropriate to determine, using the CAPM, a discount rate for each segment150 and to discount to present value, using the DCF model, the estimated free cash flows and terminal value of each line of business. In the words of Brealey and Myers, “The true cost of capital depends on the use to which the capital is put.”151

This is basically the method that was followed in valuing the different business segments of Warner Brothers in the acquisition by Time of Warner Brothers to form Time-Warner. The investment banking firms of Wasserstein/Shearson, in their opinion to Time’s board,152 and Lazard in its opinion to Warner’s board,153 used a range of discount rates for each of Warner’s business segments. Neither report indicates, however, whether the discount rates were determined by using CAPM.

Some companies use different discount rates depending on the type of investment being analyzed. For example, an investment that expands the firm’s core business may be analyzed with a discount rate equal to the firm’s cost of capital, determined using WACC. A more risky venture may be analyzed with a higher discount rate, and a less risky venture may have a lower rate.154

CAPM was used by the Delaware Court of Chancery in the appraisal decision in Cede & Co. v. Technicolor, Inc.,155 where the court said, “The CAPM methodology is certainly one of the principal [sic] ‘techniques or methods . . . generally considered acceptable [for estimating the cost of equity capital component of a discounted cash flow modeling] in the financial community. . . .’”156

IX. LAZARD’S USE OF BETA, CAPM, AND THE FORMULA FOR LEVERING AND DELEVERING BETA IN COMPUTING CONTRAIL’S WACC AND PRESENT VALUE

A. Introduction

Lazard Frères & Co., LLC represented Conrail in the proposed acquisition of Conrail by competing acquirors CSX and Norfolk Southern . Appendix A presents Lazard’s use of levered and unlevered betas and CAPM in computing an estimate of Conrail’s WACC, and Appendix B shows Lazard’s discounted cash flow valuation of Conrail’s cash flows, with the discount rate determined by the use of WACC as indicated in Appendix A. This Part examines these two appendices in detail.

150. Thompson, supra note 1, at 513 (citing BREALEY & MYERS, supra note 19, at 189). 151. Thompson, supra note 1, at 513 (citing BREALEY & MYERS, supra note 19, at 182). 152. Thompson, supra note 1, at 513 (directing the reader to Appendix C, at C-2 to C-3 of that article). 153. Thompson, supra note 1, at 513 (referencing Appendix D, at D-5 to D-6 of that article). 154. Thompson, supra note 1, at 513 (citing BREALEY & MYERS, supra note 19, at 189). 155. No. CIV.A. 7129, 1990 WL 161084 (Del. Ch. Oct. 19, 1990). 156. Thompson, supra note 1, at 513 (quoting Cede & Co., 1990 WL 161084, at *28, which cited Weinberger v. UOP, Inc., 457 A.2d 701, 713 (Del. 1983), and referencing Northern Trust Co. v. C.I.R., 87 T.C. 349, 368 (1986)).


B. Computation of Levered Beta

Lazard’s starting point was the computation of Conrail’s levered beta. Rather than focusing only on Conrail, Lazard uses the levered betas of five railroad firms (Burlington Northern, CSX, Conrail, Norfolk Southern, and Union Pacific)157 to compute an average levered beta.158 This approach is consistent with the recommendation of Brigham and Gapenski, Brealey and Myers, and Copeland, Koller, and Murrin, all of who say that a single firm’s beta may be unstable and, therefore, recommend using a portfolio of firms in a particular industry in estimating a firm’s beta.159 Here the average levered beta of 0.88 is close to Conrail’s levered beta of 0.85.160

Appendix A does not indicate how Lazard determined the levered betas for the firms. The estimates could have come from a beta book,161 from a service like Bloomberg,162 from regressing the returns on these stocks against the returns on the market,163 or from use of the covariance statistical technique.164 If the regression or covariance technique is used, weekly or daily returns for the market and the stocks should be used for a period of at least five years.165 The result here is the levered beta because the computation was done with respect to the equity of each firm, thereby automatically taking into account the firm’s leverage.

C. Conversion of Levered Beta to Unlevered Beta

The next step used by Lazard was to compute the unlevered betas for each firm and to determine an average unlevered beta.166 Although Appendix A does not indicate what method Lazard used in converting the levered betas in column [3] of Appendix A into unlevered betas in column [4] of Appendix A, it is clear that it used the delevering formula set out in Part VII:

T)(D/E)]-(1 [11

+= ββu

Here βu is the unlevered beta; β l is the levered beta; T is the corporate income tax rate, which, as indicated in chart [8] of Appendix A, is 40%; D is the debt set out in column [6] of Appendix A; and E is the equity set out in column [7] of Appendix A.

The following table demonstrates that this formula converts the levered betas in column [3] of Appendix A into the unlevered betas in column [4] of Appendix A:

157. See infra Appendix A[2] for Lazard’s examination of these five railroad firms. 158. See infra Appendix A[2], A[3]. 159. See supra Part I.C. 160. See infra Appendix A[3]. 161. See supra Part I.B. 162. See supra Part I.B. 163. See supra Parts IV and V. 164. See supra Part III. 165. See supra Part II.C. 166. See infra Appendix A[4].


Table RR Conversion of Levered Betas to Unlevered Betas

[1]

(From Appendix A[2])

[2] (From

Appendix A[3])

[3] [4] (Compare Appendix

A[4]) Comparable Company

Levered Beta

÷ Delevering Formula Unlevered Beta

1. Burlington Northern

0.95 ÷ 1 + (1 -T)(D/E)*

0.95 ÷ 1 + (1 - .4)(4.2/12.8) 0.95 ÷ 1 + (.6)(.33) 0.95 ÷ 1 + .19 0.95 ÷ 1.19 = 0.79

2. CSX 0.98 ÷ 1 + (1-T)(D/E) 0.98 ÷ 1 + (1 - .4)(2.8/10.5) 0.98 ÷ 1 + .6 (.27) 0.98 ÷ 1 + .16 0.98 ÷ 1.16 = 0.84

3. Conrail 0.85 ÷ 1 + (1-T)(D/E) 0.85 ÷ 1 + (.6)(2.0/5.7) 0.85 ÷ 1 + (.6)(.35) 0.85 ÷ 1 + .21 0.85 ÷ 1.21 = 0.70

4. Norfolk Southern 0.82 ÷ 1 + (1+T)(D/E) 0.82 ÷ 1 + (.6)(1.8/11.6) 0.82 ÷ 1 + (.6)(.15) 0.82 ÷ 1 + .09 0.82 ÷ 1.09 = 0.75

5. Union Pacific 0.82 ÷ 1 + (1-T)(D/E) 0.82 ÷ 1 + (.6)(6.1/10.2) 0.82 ÷ 1 + (.6)(.59) 0.82 ÷ 1 + (.35) 0.82 ÷ 1.35 = 0.60 Avg. 0.74

* The estimate of Debt (D) and Equity (E) comes from Appendix A[6] and A[7].


D. Recomputation of Levered Beta for Various Debt to Equity Ratios

Lazard’s next step was to recompute a levered beta for Conrail under the various debt to equity financing assumptions set out in columns [9] [a] and [b] of Appendix A, as follows:

APPENDIX A[9][a] and [b] [a] [b] Debt/Capital Debt/Equity

1. 0.0% 0.0% 2. 10.0% 11.1% 3. 20.0% 25.0% 4. 30.0% 42.7% 5. 40.0% 66.7% 6. 50.0% 100.0%

Thus, Lazard decided to determine a levered beta for Conrail under the assumption that Conrail’s capital structure included from zero to 50% debt. Notice that the ratio of debt to total capital (including debt and equity) in column [9][a] is less than the corresponding debt to equity ratio in column [9][b].

Column [9][c] has an average unlevered beta of 0.74, which was computed in column [4], as discussed above. Column [9][d] sets out a levering factor that is applied to the average unlevered beta in column [9][c] to come up with a levered beta in column 9[e]. Although Appendix A does not indicate how Lazard came up with the levering factor, it is clear that Lazard used the following formula for determining a levered beta, which is set out in Part VII:

β l = βu[1 +(1-T)(D/E)]

The levering factor here is computed by the following formula: [1 +(1-T)(D/E)]. The T in this formula is the 40% tax rate indicated in column [8][a], the D/E term is set out in percentage form in column [9][b], which is derived from the debt to capital ratio in column [9][a]. For example, if the debt in column [9][a] is 20%, the equity is 80%, and, as indicated in column [9][b], the ratio of debt to equity expressed as a percentage in column [9][b] is 25%. Thus, the levering factor set in column [9][d] is computed as follows:


Table SS Computation of Levering Factors

Debt:Equity Debt/Equity as a %

(See column [9][b])

Levering Formula [1 +)1 - T)(D/E)]

=

Levering Factor (See column

[9][d]) 1. 0%:100% 0.0% [1 + (.60)(0.0)] = 1 2. 10.0%:90.0% 11.1% [1 + (.60)(.111)] = 1.07 3. 20.0%:80.0% 25.0% [1 + (.60)(.250)] = 1.15 4. 30.0%:70.0% 42.9% [1 + (.60)(.429)] = 1.26 5. 40.0%:60.0% 66.7% [1 + (.60)(.667)] = 1.40 6. 50.0%:50.0% 100% [1 + (.60)(1.00)] = 1.60

With the above levering factors, Lazard then computed the levered beta for the

various debt to equity ratios as follows:

Table TT Computation of Levered Betas

Debt:Equity Debt/Equity As a %

(See column [9][b])

Average Unlevered

Beta (See column

[9][c])

x

Levering Factor (See

column [9][d])

=

Levered Beta

(See column [9][e])

1. 0%:100% 0.0% 0.74 x 1.00 = 0.74 2. 10.0%:90% 11.1% 0.74 x 1.07 = 0.79 3. 20.0%:80% 25.0% 0.74 x 1.15 = 0.85 4. 30.0%:70% 42.9% 0.74 x 1.26 = 0.93 5. 40.0%:60% 66.7% 0.74 x 1.40 = 1.03 6. 50.0%:50% 100% 0.74 x 1.60 = 1.18

E. Use of CAPM to Compute Cost of Levered Equity

Lazard’s next step was to use the estimate of the levered beta under these six possible capital structures to compute a cost of levered equity under each structure. Column [9][f] sets forth the cost of equity, which is the cost of levered equity, and this is the result of the use of CAPM. CAPM provides that the cost of levered equity is equal to the risk-free rate plus the beta multiplied by the market risk premium, which is set forth as follows:

rle = rf + β l(rm - rf)

Here rle is the cost of levered equity, rf is the risk-free rate, β l is the beta for levered equity, and (rm-rf) is the return on the market minus the risk-free return, or the market risk premium.

Column [8][b] of Appendix A gives the risk-free rate (rf) as 6.64%. This is presumably the T-bond rate. Column [8][c] gives the market risk premium (rm-rf) as 7.0%, and column [9][e] gives the levered beta under the six capital structures. Thus, the


cost of equity in column [9][f], which is the cost of levered equity (rle), was computed by Lazard using CAPM as follows:

Table UU Lazard’s Computation of Conrail’s Cost of Levered Equity Using CAPM

Capital Structure

Debt:Equity

Cost of Levered Equity

rle

=

Risk-Free

Rate (See

column [8][b])

+

Levered β x Market Risk

Premium (See columns [9][e]

& [8][c])

=

Cost of Equity (See

column [9][f])

1. 0% 100% rle = 6.64% + [(0.74) x (7.0%] = 11.8% 2. 10.0% 90.0% rle = 6.64% + [(0.79) x (7.0%)] = 12.1% 3. 20.0% 80.0% rle = 6.64% + [(0.85) x (7.0%)] = 12.6% 4. 30.0% 70.0% rle = 6.64% + [(0.93) x (7.0%)] = 13.1% 5. 40.0% 60.0% rle = 6.64% + [(1.03) x (7.0%)] = 13.9% 6. 50.0% 50.0% rle = 6.64% + [(1.18) x (7.0%)] = 14.9%

Thus, Table UU above demonstrates that the cost of equity, as determined by

CAPM, increases from 11.8% when zero debt is used in the capital structure (that is, the cost of unlevered equity) to 14.9% when the capital structure contains 50% debt. This illustrates that 11.8% is the cost of equity with only business risk and that this cost increases as the financial risk to the firm increases.167

F. Computation of Conrail’s Weighted Average Cost of Capital

Column [11] of Appendix A shows the computation of Conrail’s weighted average cost of capital (WACC) under the six different capital structures and under seven different assumptions about the cost of debt, as set forth in column [10]. For illustrative purposes we will assume that the pre-tax cost of debt for Conrail is 10% and the after-tax cost is 6%, as indicated in column [10][g]. Since the tax rate is 40% as indicated in column [8][a], the after-tax cost of debt must be 6.0% when the cost of pre-tax debt is 10%, computed as follows:

After-Tax Cost of Debt = Pre-Tax Cost of Debt x (1-T) = 10% x (1-0.40) = 10% x 0.60 = 6%

Lazard computed the WACCs under the six capital structures by applying the

applicable weight for the debt and equity under each of the cases pursuant to the following formula:

167. See supra Part VII.


WACC Formula WACC = [(Percentage of Debt (D) in Capital Structure (D+E)) x

(After-tax Cost of Debt) + (Percentage of Equity (E) in Capital Structure (D+E)) x (Cost of Levered Equity)]

WACC

=

+ EDD x (Pre-tax Cost of Debt (rd)) (1 – Tax Rate (T))

+

+ EDE x (cost of Levered Equity (rle))

WACC

=

+ EDD x (rd)(1 – T) +

+ EDE x rle

In Table VV below,

+ ED

D is set out in column [2], (rd)(1 – T) is set out in column

[3], the result of

+ ED

D x (rd)(1-T) is set out in column [4],

+ ED

E is set out in column

[5], rle is set out in column [6], the result of lerED

E×

+

is set out in column [7], and the

WACC is set out in column [8].

Table VV (Part 1) Computation of Conrail’s WACC

[1]

Debt: Equity

[2] Percentage

Debt in Capital

Structure

+ ED

D

x

[3] After Tax

Cost of Debt (rd(1-T))

=

[4] Weighted

Average Cost of Debt

1 0% : 100% 0% x 6.0% = 0% 2 10.0% : 90.0% 10% x 6.0% = 0.6% 3 20.0% : 80.0% 20% x 6.0% = 1.2% 4 30.0% : 70.0% 30% x 6.0% = 1.8% 5 40.0% : 60.0% 40% x 6.0% = 2.4% 6 50.0% : 50.0% 50% x 6.0% = 3.0%


Table VV (Part 2)

Computation of Conrail’s WACC [5]

[Percentage of Equity in

Capital Structure

+ ED

E

x

[6] Cost of Equity

(See column [9][f])

=

[7] Weighted Average Cost of Equity

[8] WACC

[Column [4] plus column [7]]

(See Appendix A column [11][g])

1. 100% x 11.8% = 11.80% 11.8% 2. 90% x 12.1% = 10.89% 11.5% 3. 80% x 12.6% = 10.08% 11.3% 4. 70% x 13.1% = 9.17% 11.0% 5. 60% x 13.9% = 8.34% 10.7% 6. 50% x 14.9% = 7.45% 10.5% As indicated in column [8] of Table VV above, the WACC goes down as the use of

debt in the capital structure increases. Brigham and Ga penski point out that, at some point, the additional use of debt should cause the weighted average cost of capital to begin to rise because “financial distress and agency costs become increasingly important, offsetting some of the tax advantages [of using debt].”168

G. Use of WACC in Determining the Present Value of Conrail’s Free Cash Flow

The next step followed by Lazard was to use the estimate of Conrail’s WACC set out in Appendix A as the discount rate in the Discounted Cash Flow analysis in Appendix B. As seen in column [3] of Appendix B, Lazard used the following three estimates of Conrail’s WACC as the discount rate: 11.0%, 11.5%, and 12.0%. It is standard practice to use a range of discount rates as part of a sensitivity analysis.169 On this point Judson P. Reis and Charles R. Cory in a chapter entitled the Fine Art of Valuation, in the Mergers and Acquisitions Handbook write:

As a practical matter . . . the weighted average cost of capital never is used as a point estimate of the ‘right’ discount rate. Instead, this calculation defines the center of a range of discount rates (usually one or two percentage points on either side of the estimate) that will be used to discount the cash flows.170 In this case, the 11.5% midpoint of the range in Appendix B[3] is the same as the

estimate of Conrail’s WACC with 10% debt in the capital structure and a 10% pre-tax cost of debt, as demonstrated in Appendix A[11][g]. Because approximately 25% of Conrail’s capital is in debt as demonstrated in Appendix A[6] and [7], the WACC with a

168. BRIGHAM & GAPENSKI, supra note 1, at 639-40. 169. See Thompson, supra note 1, at 499-500. 170. Judson P. Reis & Charles R. Cory, The Fine Art of Valuation, in ROCK ET AL., supra note 1, at 184.


25% debt and a 10% pre-tax interest rate would have been 11.2%, determined by extrapolation from Appendix A [11][g].

Although we do not know exactly how Lazard arrived at the 11.0%, 11.5%, and 12.0% estimates of the discount rate used in Appendix B[3], the assumptions appear to be realistic given the estimates of WACCs computed in Appendix A[11].

These three discount rates were used to compute the present value of the free cash flows computed in Table [1] of Appendix B. This table computes the free cash flows by starting with an estimate of Conrail’s net income for each year from 1997 through 2003 as indicated in Table [1][a] of Appendix B and adding back the after-tax interest expense shown in Table [1][b] to compute unlevered net income (Table [1][c]), that is, income computed without any interest expense. In explaining the rationale for this adjustment, Reis and Cory explain:

DCF valuations use a discount rate that reflects the firm’s weighted-average cost of capital or the price it must pay to suppliers of both debt and equity. Accordingly, the free cash flows to be discounted should be developed independent of financing costs. In valuing a going concern with existing liabilities, therefore, the after tax cost of interest is added back to [net income] to create an unlevered free cash flow.171 Additionally, the following adjustments are made in computing free cash flow:

depreciation and amortization (Table [1][d]) are added back, capital expenditures (Table [1][e]) are deducted and the adjustments relating to working capital (Table [1][f]), deferred taxes (Table [1][g]), and other items (Table [1][h]) are made in computing the free cash flow set out in Table [1][i]. The result here is consistent with the standard method of computing the free cash flows.172 The present value of these free cash flows under the three discount rates is set out in Column [4] of Appendix B, which is labeled NPV of Cash Flows.173

The terminal value of Conrail for year 2004, the year after the last year for the cash flow projections, is estimated by reference to Conrail’s earnings before interest, taxes, depreciation, and amortization (EBITDA). Conrail’s EBITDA is set forth on Line[2] of Appendix B, but the EBITDA for the year 2004 is not given. In any event, Table [5] of Appendix B shows that a range of terminal values is estimated by multiplying the EBITDA for year 2004 by 8, 8.5, and 9. These multiples were apparently determined by considering multiples of comparable companies or comparable transactions. This is a standard way for determining the terminal values,174 and it is also appropriate to use a range of estimates of terminal value as part of a sensitivity analysis.175 Thus, Table [5] shows the present value of the range of terminal values determined using the range of discount rates.

171. Id. at 187. 172. See Thompson, supra note 1, at 481-89; Reis & Cory, supra note 170, at 182. 173. Technically, column [4] presents the present values of the free cash flows, not the net present value (NPV). NPV is the difference between (1) the cost of an investment and (2) the present value of the free cash flows and terminal value. 174. Thompson, supra note 1, at 489-98. 175. Id. at 499-500.


Table [6] of Appendix B presents the range of discounted present values of Conrail’s free cash flows and terminal values, thus giving the value of Conrail’s assets (i.e., Conrail’s enterprise value). This table is a summation of the present values of the cash flows in Table [4] and the present values of the terminal values in Table [5].

Table [8] shows the estimate of the range of Conrail’s equity value by subtracting from the enterprise value in Table [6], Conrail’s net debt (i.e., debt minus cash) of $2.102 billion set out in Table [7]. Thus, for example, using the 11% discount rate, the range of equity values of Conrail go from $7.271 billion, with a terminal value computed at 8 times the 2004 EBITDA, to $8.125 billion, with the terminal value computed at 9 times the 2004 EBITDA.

Table [10] shows the equity value on a per share basis after taking account of certain option proceeds, which were not valued using the discounted cash flow technique. Thus, Table [10] shows that the value per share of Conrail ranged from a low of $74.68, with a discount rate of 12% and the terminal value determined by using a multiple of 8 times the 2004 EBITDA, to a high of $89.51, with a discount rate of 11%, and the terminal value determined by using a multiple of 9 times the 2004 EBITDA. This valuation turned out to be significantly less than the $115 per share price finally paid by Norfolk Southern.176

X. SUMMARY OF LAZARD’S DCF VALUATION APPROACH

Lazard used beta and CAPM to estimate Conrail’s cost of levered equity under various capital structures, and then estimated Conrail’s WACC under various capital structures with various costs of debt. Lazard then, exercising its judgement based on the WACCs in Table [11] of Appendix A, used rates of 11%, 11.5%, and 12% as a range of discount rates for determining the present value of Conrail’s free cash flows and terminal value. By using WACC in determining the discount rate, Lazard automatically took into account the financing effect and, therefore, it was appropriate for Lazard to discount Conrail’s free cash flows without deducting any interest payments. The present value determined by discounting the free cash flows and the terminal value is the estimated value of Conrail’s assets . Lazard, therefore, deducted Conrail’s net debt in computing the fair market value of Conrail’s equity.

As an alternative, Lazard could have used the adjusted net present value (APV) method suggested by Brealey & Myers.177 With APV, Lazard first would have discounted Conrail’s free cash flows (without interest deduction) and terminal value at Conrail’s cost of unlevered equity, and then discounted the expected tax savings from the interest deduction at the same cost of unlevered equity. The sum of the two present values would have been the value of Conrail’s assets. Lazard would have then deducted Conrail’s net debt in computing the fair market value of Conrail’s equity.

As another option, Lazard could have used the equity DCF approach suggested by Brigham and Gapenski.178 Under this approach, Lazard would first have computed Conrail’s WACC using the levered equity beta as it did in Appendix A. Lazard would then have discounted Conrail’s free cash flows with interest deducted and its terminal

176. See Agis Salpukas, Conrail Chugs off into the Sunset, N.Y. T IMES, June 1, 1999, at C1. 177. See supra Part I.B. 178. Id.


value at the WACC. The interest is deducted in this approach, because the equity is being valued.179 The result would be the fair market value of Conrail’s equity.

XI. CONCLUSION

The discounted cash flow technique is commonly utilized by firms in determining whether to make an investment in a capital project, including in determining the valuation of a target in a merger or acquisition transaction. The determination of the appropriate cost of capital is a necessary element in the discounted cash flow model. The most commonly used technique for determining the cost of equity capital is the CAPM, which posits that the cost of equity capital is equal to the sum of (1) the risk-free rate and (2) the beta for the investment times the market risk premium. Thus, beta is an essential element in CAPM and, therefore, is often an essential element in the discounted cash flow model.

Beta, which is a measure of the sensitivity of the returns for a particular stock to the returns on the market, can be derived through the use of the concepts of covariance and variance, and through the use of regression analysis, including Excel’s linear regression function. Betas for a target firm can be expressed either as levered or unlevered, and depending on the circumstances, it may be necessary to move from a levered to an unlevered beta, or vice versa. A formula developed by Robert Humada permits movements between levered and unlevered betas.

The analysis in Parts VI and VII of Lazard’s valuation of Conrail, which was the target of competing bids by CSX and Norfolk Southern, illustrates in a real life acquisition context the computation of levered and unlevered betas and their use in a discounted cash flow model.

179. BRIGHAM & GAPENSKI, supra note 1, at 1089.


APPENDIX A

Lazard’s Computation of Conrail’s Weighted Cost of Capital

[1] Ticker

[2] Comparable Company

[3] Levered

Beta

[4] Unlevered

Beta

[5] Debt Cap.

Ratio

[6] Debt

[7] Equity Value

BNI Burlington Northern 0.95 0.79 24.8% $4,239.0 $12,834.6

CSX CSX 0.98 0.84 21.3% 2,848.0 10,522.7

CRR Conrail 0.85 0.70 26.5% 2,078.0 5,755.7

NSC Norfolk Southern 0.82 0.75 13.1% 1,765.0 11,658.7

UNP Union Pacific 0.82 0.60 37.6% 6,155.0 10,223.0

AVERAGES 0.88 0.74 24.7% $3,417.0 $10,199.0

[8] Assumptions

[a] Marginal Tax Rate: 40.00% [b] Risk Free Rate of Return: 6.64% [c] Market Risk Premium: 7.00%

[9] Computation of Cost of Equity

[a]

[b]

[c]

[d]

[e]

[f]

Debt/Cap.

Debt/Equity

Average Unlevered Beta

Levering Factor

Levered Beta

Cost of Equity

0.0%

0.0%

0.74

1.00

0.74

11.8%

10.0%

11.1%

0.74

1.07

0.79

12.1%

20.0%

25.0%

0.74

1.15

0.85

12.6%

30.0%

42.9%

0.74

1.26

0.93

13.1%

40.0%

66.7%

0.74

1.40

1.03

13.9%

50.0%

100.0%

0.74

1.60

1.18

14.9%

[10]

Pre-Tax/After-Tax Cost of Debt [a] [b] [c] [d] [e] [f] [g] [h]

7.00% 7.50% 8.00% 8.50% 9.00% 9.50% 10.00% 10.50%

4.20% 4.50% 4.80% 5.10% 5.40% 5.70% 6.00% 6.30%


[11] Weighted Average Cost of Capital

After-Tax Cost of Debt (See [10])

Debt/Cap. (See [9a]) [a]

4.20% [b]

4.50% [c]

4.80% [d]

5.10% [e]

5.40% [f]

5.70% [g]

6.00% [h]

6.30% 0.0% 11.8% 11.8% 11.8% 11.8% 11.8% 11.8% 11.8% 11.8%

10.0% 11.4% 11.4% 11.4% 11.4% 11.5% 11.5% 11.5% 11.6%

20.0% 10.9% 11.0% 11.0% 11.1% 11.1% 11.2% 11.3% 11.3%

30.0% 10.5% 10.5% 10.6% 10.7% 10.8% 10.9% 11.0% 11.1%

40.0% 10.1% 10.1% 10.2% 10.4% 10.5% 10.6% 10.7% 10.8%

50.0% 9.6% 9.7% 9.9% 10.0% 10.2% 10.3% 10.5% 10.6%

APPENDIX B Lazard’s Discounted Cash Flow Analysis of Conrail

[1]

Projected for the Year Ended December 31 ($ in Millions)

1997 1998 1999 2000 2001 2002 2003

[a] Net Income $ 488 $ 559 $ 630 $ 691 $ 739 $ 811 $ 885

[b] Plus: After-Tax Interest 96 85 72 56 56 34 11

[c] Unlevered Net Income 584 644 702 747 795 845 897

[d] Depreciation & Amortization 282 290 302 311 320 329 339

[e] Capital Expenditures (510) (550) (550) (562) (578) (595) (612)

[f] Changes in Working Capital 3 (5) 1 16 16 17 17

[g] Deferred Taxes 90 101 111 96 99 102 105

[h] Other (43) (43) (43) (43) (43) (30) (30)

[i] Free Cash Flow $ 406 $ 438 $ 523 $ 566 $ 609 $ 668 $ 715

Memo Items:

[2] EBITDA $ 1,217 $ 1,321 $ 1,425 $ 1,507 $ 1,592 $ 1,681 $ 1,774


[3]

Discount Rate

[4] NPV of

Cash Flows

[5]

Terminal Value 2004 EBITDA Multiple:

[6] Enterprise Value

2004 EBITDA Multiple: 8.0x 8.5x 9.0x 8.0x 8.5x 9.0x

11.0%

$ 2,539

+

$ 6,834

$ 7,261

$ 7,688

=

$ 9,373

$ 9,800

$ 10,227

11.5%

2,494

+

6,622

7,036

7,450 =

9,117

9,530

9,944

12.0%

2,451

+

6,418

6,819

7,220 =

8,869

9,270

9,671

[7] Less Net

Debt

[8] Equity Value (b)

2004 EBITDA Multiple:

[9] Options Proceeds

[10] Equity Value Per Share 2004 EBITDA Multiple:

8.0x 8.5x 9.0x 8.0x 8.5x 9.0x $ (2,102) $ 7,271 $ 7,696 $ 8,125 $ 76 $ 80.18 $ 84.84 $ 89.51 (2,102) 7,015 7,428 7,842 76 77.38 81.90 56.42 (2,102) 6,767 7,168 7,569 76 74.68 79.06 83.44

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