Estimating Security Betas 1

Estimating Security Betas Risk and Return We know that investors demand a premium for bearing risk, i.e., the higher the riskiness of a security, the higher its expected return must be to induce investors to buy or hold. Within the context of MPT, part of the risk that can be eliminated by forming diversified portfolios is called unsystematic or diversifiable risk, while the part that cannot be eliminated is called systematic or market risk. If investors are concerned with the riskiness of their portfolios, rather than the risk of individual securities in the portfolio, how should the risk of an individual stock be measured? The CAPM provides a framework to analyze the relationship between risk and rate of return. The relevant risk of an individual stock is its contribution to the riskiness of a well-diversified portfolio.

)(: fmifi RRRREquationSML +=

The risk that remains after diversifying is market risk, which can be measured by the degree to which a stock tends to move up or down with the market as reflected in its beta coefficient. The beta for an individual asset is derived from a regression model (single index model) by regressing stock returns against market returns. imii eRRiModelIndexSingle ++= : where

i = Intercept of the regression line

i = Slope of the regression line (stock's sensitivity to market movement)

mR = Return on a proxy for the market portfolio

ie = Firm specific or unsystematic component of risk The betas are standardized around 1 (market beta = 1). Beta = 1 (average risk investment); Beta > 1 (above average risk); Beta < 1 (below average risk); Beta = 0 (risk-less investment). The Basic Steps for Beta Estimation 1. Decide on the estimation period: Most of the financial services use periods ranging from

2 to 5 years for the regression. 2. Decide on a return interval - daily, weekly, or monthly: Shorter intervals result in more

observations, but suffer from noise created by stocks not trading. 3. Estimate stock return: Return = (Price end - Price beginning + Dividends period)/Price beginning .

Alternatively, the compound rate of return for the stock can be computed as: LN(Pend/ Pbeginning).

Estimating Security Betas 2

4. Choose an index as a proxy for the market and estimate returns for the index for each interval during the estimation period. A wide variety of indexes are available including the S&P 500, NYSE, etc. Most investigators use the S&P 500 composite index (its ticker symbol is ^GSPC) as a proxy for the market portfolio because this index encompass a large proportion of the total market value of US stocks.

5. Collect price data for the stock and the selected index for the estimation period from the

Internet. The monthly price data in the examples below was obtained from the Yahoo finance site (http://finance.yahoo.com/). Other sites may be used as well. Enter the ticker symbol of the company [Walt Disney (DIS), Bristol Myers Squibb (BMY)], get the Basic quote, then look at More Info box and click on Chart. When the graph for the firm's prices appears, look down at the bottom you will see the link - Historical Quotes. Select the time period (Jan 1, 1997 - Jan 1, 2002 is used in the examples) and interval (monthly). You should have 61 monthly observations. Download the prices to your computer disk in the Excel format. Do the same for the S&P 500 using matching interval and periods. Create a separate worksheet with date, S&P Close values, and stock adjusted close prices. Now compute monthly returns or percentage changes in prices using: LN(Pend/ Pbeginning) for both the stock and the index.

6. Run regression in Excel to calculate the coefficient estimates for the single index model by regressing monthly stock returns (the Y in regression is the dependent variable) against percentage change in the index (the market is X or the independent variable). You can estimate the regression coefficients either using the appropriate Excel function keys or by using Regression facility of Data Analysis (Tools/Data Analysis/Regression). The use of Data Analysis is recommended because it produces a comprehensive regression output.

7. Evaluate the regression output with special attention to model fit (F-value, R2) and

statistical significance of the estimates. The R2 of the regression provides an estimate of the proportion of risk (variance) of a stock that can be attributed to market risk; the balance (1 - R2) can be attributable to firm specific risk.

Evaluating Performance

The intercept of the regression provides a simple measure of stock performance during the regression period relative to the CAPM. Combining CAPM and regression equations, intercept (alpha) = Rf (1 - beta) if stock did as well as expected during the regression period. If alpha is > Rf (1 - beta) then stock did better than expected. The alpha measure is also known as Jensen's alpha.

Examples

The following examples demonstrate estimation procedure for two companies, Walt Disney (DIS) and Bristol Myers Squibb (BMY), using monthly returns, S&P 500 as a proxy for market, and a five-year period. Only partial data are presented in the tables followed by regression outputs from Excel.

Estimating Security Betas 3

S&P 500 and Walt Disney Monthly Price and Return Data

January 1, 1997 to January 2, 2002

Date

S&P 500 Close

DIS Adj. Close

Return (MKT)

Return (DIS)

2-Jan-02 1130.20 21.06 -0.01570 0.01628 3-Dec-01 1148.08 20.72 0.00755 0.02245 1-Nov-01 1139.45 20.26 0.07248 0.09684 1-Oct-01 1059.78 18.39 0.01794 -0.00163 4-Sep-01 1040.94 18.42 -0.08526 -0.31182 1-Aug-01 1133.58 25.16 -0.06626 -0.03553 2-Jul-01 1211.23 26.07 -0.01080 -0.09227 1-Jun-01 1224.38 28.59 -0.02535 -0.09024 1-May-01 1255.82 31.29 0.00508 0.04444 2-Apr-01 1249.46 29.93 0.07401 0.05600 1-Mar-01 1160.33 28.30 -0.06636 -0.07879 1-Feb-01 1239.94 30.62 -0.09683 0.01613 2-Jan-01 1366.01 30.13 0.03405 0.05107

1-Jan-97

Count (number of observations) 60 60 Average Monthly Return 0.0060 -0.0017 Standard Deviation of Return 0.0517 0.0958

Regression Output: Summary Regression Statistics Multiple R 0.51748 R Square 0.26779 Adjusted R Square 0.25516 Standard Error 0.08269 Observations 60 ANOVA df SS MS F Significance F Regression 1 0.14502 0.14502 21.21179 0.00002 Residual 58 0.39654 0.00684 Total 59 0.54156 Coefficients Standard Error t Stat P-value Intercept -0.00753 0.01075 -0.70012 0.48665 Market Return (Rm) 0.95955 0.20834 4.60563 0.00002

Estimating Security Betas 4

S&P 500 and Bristol Myers Squibb Monthly Price and Return Data

January 1, 1997 to January 2, 2002

Date

S&P 500 Close

BMY Adj Close

Return (MKT)

Return (BMY)

2-Jan-02 1130.20 45.04 -0.01570 -0.11145 3-Dec-01 1148.08 50.35 0.00755 -0.05261 1-Nov-01 1139.45 53.07 0.07248 0.00567 1-Oct-01 1059.78 52.77 0.01794 -0.03391 4-Sep-01 1040.94 54.59 -0.08526 -0.01039 1-Aug-01 1133.58 55.16 -0.06626 -0.05210 2-Jul-01 1211.23 58.11 -0.01080 0.12816 1-Jun-01 1224.38 51.12 -0.02535 -0.03649 1-May-01 1255.82 53.02 0.00508 -0.03193 2-Apr-01 1249.46 54.74 0.07401 -0.05405 1-Mar-01 1160.33 57.78 -0.06636 -0.06532 1-Feb-01 1239.94 61.68 -0.09683 0.02429 2-Jan-01 1366.01 60.20 0.03405 -0.17398

1-Jan-97

Count (number of observations) 60 60 Average Monthly Return 0.0060 0.0074 Standard Deviation of Return 0.0517 0.0746

Regression Output: Summary Regression Statistics Multiple R 0.27468 R Square 0.07545 Adjusted R Square 0.05951 Standard Error 0.07238 Observations 60 ANOVA df SS MS F Significance F Regression 1 0.02480 0.02480 4.73326 0.03367 Residual 58 0.30385 0.00524 Total 59 0.32865 Coefficients Stand. Error t Stat P-value Intercept 0.00498 0.00941 0.52961 0.59840 Market Return (Rm) 0.39678 0.18238 2.17561 0.03367