Question

# I need help answering the 3 questions below.

Please see the case study about coca cola to answer the questions below:

1. Can you find an error in equation 1

2. Look at Figure 1, how well does the regression line predict the actual results? What does the R2 in the regression equation (table1) mean? Does this give you confidence in the calculated Beta or not?

3. The authors use very long-term ranges to estimate risk-free rate and market premium. What are the benefits and risks of this? Would you use a different range of numbers?

Calculating The Beta Coefficient And

Required Rate Of Return For Coca-Cola

John C. Gardner, University of New Orleans, USA

Carl B. McGowan, Jr., Norfolk State University, USA

Susan E. Moeller, Eastern Michigan University, USA

ABSTRACT

In this paper, we demonstrate how to compute the required rate of return for Coca-Cola using

modern portfolio theory with data downloaded from the internet. We demonstrate how to

calculate monthly returns for the index and Coca-Cola and how to use the returns to compute the

beta coefficient and the required rate of return using the downloaded data. We show how to

validate the data for the market index and the company and how to compute the returns using the

dividend and stock split adjusted prices. We demonstrate how to graph the characteristic line for

Coca-Cola and use the graph to check that the regression was run correctly. We use Coca-Cola

and the S&P 500 Index in this paper, but any company listed on Yahoo! Finance can be used as

the example. This paper can be used as the basis of a lecture on intermediate corporate finance

or investments to demonstrate the process using a real company-

Keywords: beta; characteristic line; required rate of return; Coca-Cola; teaching note

INTRODUCTION

arkowitz' (1952) began modern portfolio theory (MPT) which can be used to explain the

relationship between risk and return for assets, particularly stocks. Stock of companies that have

higher rates of return have higher levels of risk. In order to achieve a lower level of risk, an

investor must accept a lower expected rate of return. This concept is called the dominance principle and allows for

the creation of the efficient frontier. MPT partitions risk into non-systematic risk, which can be eliminated from a

portfolio through diversification, and systematic risk that is market wide and cannot be diversified. Non-systematic

risk is company specific and is reduced to zero in a large, well diversified portfolio. In order to determine

systematic risk for a stock, we use the market model developed by Sharpe (1964). The returns for a stock are

regressed as the dependent variable against a market index used as the independent variable. The slope coefficient

of the regression is the measure of systematic risk for the stock. Systematic risk measures the degree to which a

stock moves with the market. A higher beta coefficient implies that returns for the stock move more than the market

and a lower beta coefficient implies that returns for the stock move less that the market. The former are aggressive

stocks and the latter are defensive stocks.

In this paper, we show how to retrieve data from the internet, how to compute returns for both the market

index and the stock, and how to run a regression to determine the beta coefficient to measure the systematic risk for

the stock. In addition, we show how to graph the data with a trend line and statistics to verify that the first

regression is run correctly; that is, with the correct variable as the independent variable. We show how to do all of

this analysis using Excel.

DOWNLOADING DATA FROM THE INTERNET

The data used for the analysis discussed in this paper are downloaded from the internet using the Yahoo!

Finance website. The URL for Yahoo! Finance is http///finance.yahoo.com/. Once one arrives at the Yahoo!

Finance website, the S&P 500 data can be found by clicking on the "S&P500" icon and then, clicking on the

Historical Prices" icon. Click on the "Monthly" indicator to download monthly data and enter the dates. For this

paper we download sixty-one monthly, observations in order to calculate sixty monthly returns. The data columns

are: Date, Open, High, Low, Close, Average Volume, and Adjusted Close. The index and the Coca-Cola price are

adjusted for splits and dividends. Move the cursor to the bottom of the data and click on "Download to

Spreadsheet". Save the data to a spreadsheet and repeat the process for the Coca-Cola data. Begin by entering the

Coca-cola ticker symbol, KO, and download and save the data for the save time period.

CALCULATING RETURNS FOR THE S& P 500 INDEX AND FOR COCA-COLA3

In this paper, we use arithmetic returns to compute the beta coefficient for Coca-Cola. Arithmetic returns

are calculated by dividing the ending index or stock value, (Value,), by the beginning value, (Value,), and

subtracting one as in Equation [1]. An alternative method to calculate the return is to subtract the beginning value,

(Value,), from the ending value, (Value,), and dividing by the beginning value, (Value,), as in Equation [2]. Both

returns are adjusted for dividends and stock splits. The returns used in the regression analysis are arithmetic returns.

Return = [(Value,- Valuco) - 1]

Retum = [(Value,- Value.)Valuco)]

[1]

[2]

Five years of Monthly data are used to generate sixty data points".

CALCULATING BETA FOR COCA-COLA

Modern Portfolio Theory shows that investors are rewarded for the systematic risk of an investment and not

for the total risk of an investment because total risk includes firm specific risk that can be eliminated in a well

diversified portfolio. The specific risk of an individual stock is the slope coefficient of the characteristic line which

is the regression line between the monthly returns for the individual security and the monthly returns for the market

index. Beta coefficient lines are calculated using a sixty month regression. In this example, the beta coefficient for

Coca-Cola is calculated using sixty monthly observations of returns for Coca-Cola from 09/02/2003 to 08/01/2008

and returns for the S&P 500 Index for the same time period. Beta is the covariance between returns for Coca-Cola

and returns for the S&P 500 divided by the variance for the S&P 500.

Rko = Alphago + Betako (R.)

[3]

Rko

the return for Coca-Cola stock

Betaxo

the slope of the regression line between returns for the market and returns for Coca-Cola

Alphako

the intercept coefficient for the regression line between returns for the market and returns for Coca-

Cola

(R.)

the return on the S&P 500 Stock market Index

(Rm - RF)

the market risk premium is the additional return that stock holders receive for the additional risk of

holding stocks rather than the risk free asset, long-term government bonds.

Appendix A contains the data used to compute the Coca-cola beta and are downloaded from Yahoo!

Finance. Column 1 shows the date and Columns 2 and 3 contain the stock split and dividend adjusted index and

price values, for the S&P 500 Index and for Coca-cola stock, respectively. The independent variable is the return for

the S&P500 (Column 4) and the dependent variable is the return for Coca-Cola (Column 5). The returns are

calculated by dividing the ending index or stock value by the beginning value and subtracting one. An alternative

method to calculate the return is to subtract the beginning value from the ending value and dividing by the beginning

value. Both returns are adjusted for dividend and stock splits. The returns used are arithmetic returns.

Table I contains the regression results for the regression between the return for the S&P500 and for Coca-

Cola using Excel. The independent variable is the return for the S&P500 (x-axis) and the dependent variable is the

return for Coca-Cola (y-axis). Both returns are adjusted for dividends and stock splits. The adjusted R" for the

regression is 0.23 and the F-statistic is 18.65, both of which are statistically significant at the 0.0000 level. The

regression coefficient is 0.7560 and has a t-statistic of 4.31 and is significant at the 0.0000 level.

Table 1: Coca-Cola versus the S&P 500 Regression of Arithmetic Means Returns from 09/02/03 to 08/01/08

Regression Statistics

Multiple F

0.493288

R Square

0.243333

Adjusted R Square

0.230287

Standard Error

3.723303

Observations

60

ANOVA

d'f

SS

MS

F

Significance F

Regression

1

258.57

258.57

18.65

790000 0

Residual

58

804.05

13.86

Total

59

1062.63

Coefficients

Standard Error

I Star

P-value

Intercept

0.087056

0.485744

0.179223

0.858388

X Variable I

0.765019

0.177137

4.318793

0.000062

Coca-Cola Characteristic Line (9/2/03 to 8/0 1/08)

40

y = 0.765x + 0.0871

R- = 0.2433

Coca-Cola

15

-10

-5

10

S&P 500

Figure 1: Characteristic Line - Coca-Cola

Journal of Business Case Studies - November December 2010

Volume 6 Number 6

Figure 1 is a graph of the data used to compute the Coca-cola beta, which is the characteristic line for

Coca-Cola. Figure I was created in Excel using the Chart function. The independent variable is the return for the

S&P500 (x-axis) and the dependent variable is the return for Coca-Cola (y-axis). Both returns are adjusted for

dividends and for stock splits. The chart contains the trend line and R". The statistics in the graph are the same as

the regression statistics in Table 1. The pedagogical purpose of the graph is to chart the characteristic line for Coca-

Cola and to confirm that the regression was run with the correct independent and dependent variable. If the trend

line and statistics in the graph are not identical to the numbers in the regression, the student has reversed the

variables.

CALCULATING THE REQUIRED RATE OF RETURN FOR STOCKS

Graham and Harvey (2002) find that 73.5 percent of respondents to their survey indicate that the company

of the survey respondent uses the capital asset pricing model (CAPM) to determine the component cost of common

stock equity capital. In this paper, we use the CAPM to compute the required rate of return for Coca-Cola. The

required rate of return for Coca-Cola is the minimum rate of return demanded by stockholders of Coca-Cola stock.

The model used in this paper is based on the CAPM derived from the work of Sharpe (1964).

Rxo = Ry + Betako (R - RF)

[4]

Rko

- the required rate of return for Coca-Cola Stock

Rr

- the risk free rate of return

Betako

= the beta coefficient for Coca-Cola

Rm

= the rate of return on the stock market

(Rm - RF)

= the market risk premium

The required rate of return for Coca-Cola is the risk-free rate of return plus the risk premium for Coca-

Cola. The risk premium is the beta for Coca-Cola time the market price of risk.

COMPUTING THE REQUIRED RATE OF RETURN FOR COCA-COLA (KO) USING THE CAPM

The risk free rate is the total return (income plus capital appreciation) on Long-term Government Bonds

taken from SBBI 2007 . For the years from 1926 to 1976, SBBI uses the Government Bond File from the Center for

Research in Security Prices. For the period from 1976 to 2006, the returns in SBBI 2007 are computed from data

taken from the Wall Street Journal. The yield for the bond is the discount rate that equates the expected future cash

flows, coupon payments and maturity value, to the current price. Table 2 contains a summary of the input data and

sources of that data.

Table 2: Input Data and Sources

Variable

Value

Source

Betago

0.7650

Computed

Rf

0.0580

SBBI, 2007, page 31

0.1230

SBBI, 2007, page 31

K

0.1077

Computed

We use the security market line to compute the required rate of return for Coca-Cola. We use the long-

term bond rate taken from SBBI (2007) which equals 5.8% and the long-term market return of 12.3%. The market

risk premium is 6.5%. This yields a cost of equity for Coca-Cola of 10.77%.

Rko = Rr+ Betako (Rm - RF)

[5]

10.77% = 5.8% + 0.7650 (12.3% - 5.8%)

10.77% = 5.8% +0.7650 (6.5%)

10.77% = 5.8% + 4.97%

The required rate of return for Coca-Cola stock is 10.77%.

SUMMARY AND CONCLUSIONS

In this paper, we demonstrate how to compute the required rate of return for Coca-Cola using modern

portfolio theory. Data is downloaded from Yahoo! Finance for both Coca-Cola and for the S&P 500 Index. The

adjusted stock price for Coca-Cola and the S&P 500 Index are used to compute a five-year, monthly series of

returns. The characteristic line is the regression line from the regression in which the monthly returns for the S& P

500 Index are the independent variables and the monthly returns for Coca-Cola are the dependent variables. The

regression is run using the Data Analysis Tool Pak in Excel and the Chart function. We use SBBI 2007 data to

compute the required rate of return using the market model. We compute a required rate of return for Coca-Cola

equal to 10.77%

The objective of this paper is to demonstrate how to download the data needed to compute the required rate

of return for Coca-Cola using Modern Portfolio Theory. We demonstrate how to calculate monthly returns for the

index and Coca-Cola and how to use the returns to compute the beta coefficient and the required rate of return using

the downloaded data. We show how to validate the data for the market index and the company and how to compute

the returns using the dividend and stock split adjusted prices. We demonstrate how to graph the characteristic line

for Coca-Cola and use the graph to check that the regression was run correctly. We use Coca-Cola and the S&P 500

Index in this paper, but any company listed on Yahoo! Finance can be used as the example. This paper can be used

as the basis of a lecture on intermediate corporate finance or investments to demonstrate the process using a real

company.

AUTHOR INFORMATION

John C. Gardner is the KPMG Professor of Accounting and Director of the Global Entrepreneurship Initiative in

the Department of Accounting at the University of New Orleans. He earned his undergraduate degree in accounting

from SUNY at Albany, and MBA and Ph.D. degrees in finance from Michigan State University. Dr. Gardner has

published in leading accounting, finance and management science journals including The Accounting Review,

Journal of Accounting Research, Contemporary Accounting Research, Accounting, Organizations and Society,

Journal of Financial and Quantitative Analysis and Decision Sciences. His research interests include multi-national

corporation financial management, capital structure, and financial and forensic accounting.

Carl B. McGowan, Jr., PhD, CFA is a Professor of Finance at Norfolk State University, has a BA in International

Relations (Syracuse), an MBA in Finance (Eastern Michigan), and a PhD in Business Administration (Finance) from

Michigan State. From 2003 to 2004, he held the RHB Bank Distinguished Chair in Finance at the Universiti

Kebangsaan Malaysia and has taught in Cost Rica, Malaysia, Moscow, Saudi Arabia, and The UAE. Professor

McGowan has published in numerous journals including Applied Financial Economics, Decision Science, Financial

Practice and Education, The Financial Review, International Business and Economics Research Journal, The

International Review of Financial Analysis, The Journal of Applied Business Research, The Journal of Business

Case Studies, The Journal of Diversity Management, The Journal of Real Estate Research, Managerial Finance,

Managing Global Transitions, The Southwestern Economic Review, and Urban Studies.

Susan E. Moeller is a Professor of Finance at Eastern Michigan University since 1990. Prior to joining EMU, Dr.

Moeller taught at Northeastern University in Boston and at the University of Michigan - Flint. Her corporate

experience was with Ford Motor Company. She has published in a number of journals including, Journal of

Economic and Financial Education, Journal of Business Case Studies, Journal of Global Business, Journal of

International Finance, Journal of Financial and Strategic Decisions, Management International Review, Journal of

Applied Business Research and AAll Journal.

107

Journal of Business Case Studies - November/December 2010

Volume 6. Number 6

APPENDIX A

Date

S &P500

Rates of Return S&P500 and COCA-COLA

8/1/2008

1249.01

KO

52.07

REarin

7/1/2008

-1.45

RKO

6/2/2008

1267.38

51.50

5/1/2008

51.98

-0.99

1.11

1280.00

1400.38

57.26

-8.60

0.92

4/1/2008

58.87

1.07

-9.22

3/3/2008

1385.59

1322.70

4.75

-2.73

3.29

2/1/2008

0.60

1/2/2008

1378.53

58.46

-3.48

4.12

-0.92

12/3/2007

1/1/2007

468.36

59.00

1481.14

61.37

-6.12

62.10

-0.86

3.86

10/1/2007

4.40

1.18

055

9/4/2007

1549.38

8/1/2007

1526.75

61.76

7/2/2007

1473.99

57.47

1.48

7.46

53.78

3.58

6.86

3.20

6/1/2007

1455.27

5/1/2007

1503.35

52. 11

4/2/2007

1530.62

52.31

-3.20

52.99

-1.78

-128

3/1/2007

1482.37

1420.86

52.19

3.25

4.33

1.53

2/1/2007

1406.82

48.00

8.73

1/3/2007

1438.24

46.68

-2.18

2.83

12/1/200

47.8

48.25

1.41

-2.5

11/1/200

1418.30

1 26

10/2/2006

1400.6

1377.94

46.83

46.72

1.65

3.03

9/1/2006

1335.85

44.68

3.15

0.24

7/3/200

1303.82

2.46

457

8/1/2006

0.29

1276.66

44.50

2.13

0.70

6/1/2006

1270.20

43.02

051

5/1/2006

0.01

3.44

4/3/2006

1270.0

-2.29

3/1/2006

1310.61

44.0

41.96

3.0

1.22

4.93

0.21

2/1/2006

1294.87

1/3/2006

1280.66

41.87

1280.08

41.97

1.11

-0.24

1 2/1/2005

1248.29

41.38

0.05

1.43

1 1/1/200

40.31

2.55

2.65

10/3/2005

1249.48

1207.01

42.69

0.10

42. 78

3.52

9/1/2005

-1.77

-021

122 8.81

43.19

0.95

8/1/2005

0.69

7/1/2005

1220.33

44.00

-1.12

-1.84

055

6/1/2005

1234.18

5/2/2005

1191.33

43.76

3.60

4.81

1191.50

41.75

4/1/2005

1156.85

44.63

0.01

3.00

6.4

3/1/2005

2.74

2/1/2005

180.59

43.44

41.67

-2.01

-1.91

4.25

1/3/2005

1203.60

1181.27

41.49

1.89

2.64

12/1/2004

121 1.92

-2.53

3.16

1173.82

41.6

-036

1 1/1/200-

10/1/2004

1130.20

39.31

3.25

40.66

3.86

5.93

9/1/2004

3.3

1114.58

1104.24

40.05

1.40

8/2/200-

0.94

1.52

7/1/2004

-10.42

6/1/2004

1101.72

44.71

43.86

0.23

-3.43

1.94

5/3/2004

1140.84

4/1/200

1120.68

50.48

51.35

1.80

121

-1.69

3/1/2004

1107.30

1.54

2/2/200-

1126.21

50.57

50.30

-1.68

0.54

1/2/2004

1144.94

49.96

-1.64

1.22

0.68

12/1/2003

1.73

1.46

2.98

1 1/3/2003

111 1.92

10/1/2003

105 8.20

5.08

9.14

9/2/2003

1050.71

46.50

5.50

0.22

8/1/2003

995.97

46.40

0.71

1008.01

42.96

8.01

43.52

1.29

109

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