Unformatted text preview: WEB APPENDIX 8A
Calculating Beta Coefficients
The CAPM is an ex ante model, which means that all of the variables represent
before-the-fact expected values. In particular, the beta coefficient used in the SML
equation should reflect the expected volatility of a given stock’s return versus the
return on the market during some future period. However, people generally calculate betas using data from some past period and then assume that the stock’s
relative volatility will be the same in the future as it was in the past.
To illustrate how betas are calculated, consider Figure 8A-1. The data at the
bottom of the figure show the historical realized returns for Stock J and for the
market over the last 5 years. The data points have been plotted on the scatter
diagram, and a regression line has been drawn. If all the data points had fallen on
a straight line, as they did in Figure 8-7 in Chapter 8, it would be easy to draw an
accurate line. If they do not, as in Figure 8A-1, you must fit the line “by eye” as an
approximation or with a calculator.
Recall what the term regression line or regression equation means: The equation
Y ¼ a þ bX þ e is the standard form of a simple linear regression. It states that
the dependent variable, Y, is equal to a constant, a, plus b times X, where b is the
slope coefficient and X is the independent variable, plus an error term, e. Thus, the
rate of return on the stock during a given time period (Y) depends on what
happens to the general stock market, which is measured by X ¼ "M .
Once the data have been plotted and the regression line has been drawn
on graph paper, we can estimate its intercept and slope, the a and b values in Y ¼
a þ bX. The intercept, a, is simply the point where the line cuts the vertical axis.
The slope coefficient, b, can be estimated by the rise-over-run method. This
involves calculating the amount by which "J increases for a given increase in "M .
For example, we observe in Figure 8A-1 that "J increases from À8.9% to þ7.1% (the
rise) when "M increases from 0 to 10.0% (the run). Thus, b, the beta coefficient, can
be measured as follows:
b ¼ Beta ¼ Rise ÁY 7:1 À ( À 8:9Þ 16:0
10:0 À 0:0
10:0 Note that rise over run is a ratio. It would be the same if measured using any two
arbitrarily selected points on the line.
The regression line equation enables us to predict a rate of return for Stock J,
given a value of "M . For example, if "M ¼ 15%, we would predict that "J ¼ À8.9% þ
1.6(15%) ¼ 15.1%. However, the actual return would probably differ from the
predicted return. This deviation is the error term, eJ, for the year; and it varies
randomly from year to year depending on company-specific factors. Note, though,
that the higher the correlation coefficient, the closer the points lie to the regression
line and the smaller the errors.
In actual practice, monthly rather than annual returns are generally used for "J
and "M and 5 years of data are often employed; thus, there would be 5 Â 12 ¼ 60
data points on the scatter diagram. Also, in practice, one would use the least
squares method for finding the regression coefficients a and b. This procedure
minimizes the squared values of the error terms, and it is discussed in statistics
The least squares value of beta can be obtained quite easily with a financial
calculator. The procedures that follow explain how to find the values of beta and
the slope using a Texas Instruments or Hewlett-Packard financial calculator or a
spreadsheet program such as Microsoft Excel.
8A-1 8A-2 Web Appendix 8A FIGURE 8A-1 Calculating Beta Coefficients Historic Realized Returns
on Stock J, rJ (%)
Year 1 40 Year 5 30 _ _ rJ = aJ + b J rM + eJ
= –8.9 + 1.6rM + eJ
Year 4 7.1 –10 0 10 20 30 Historic Realized Returns
on the Market, rM (%)
∆ rJ = 8.9% + 7.1% = 16%
_ aJ = Intercept = – 8.9%
–10 _ ∆ rM = 10% ∆r
bJ = Rise = _J =
10 Year Market (rM )
" Stock (rJ )
14.9% 15.1% 26.5% _ –20
Year 2 r
sr Texas Instruments BA-II Plus
6. Press 2nd RESET ENTER to set the statistics calculation method to standard
linear regression and X, Y, and all other values to zero.
Press 2nd DATA to select the data entry portion of the calculator’s statistical
function. Once you do this, X01 appears on your screen with 0 as a value.
Key in 23.8 (the first X data point) and press ENTER to enter the first X
Press ˇ , key in 38.6, and press ENTER to enter the first Y variable.
The remaining X and Y variables may be entered by repeating Step 4.
Once all the data have been entered, press 2nd STAT to select the statistical
function desired. [LIN (stands for standard linear regression) should appear
on the calculator screen.] Then press ˇ to obtain statistics on the data. After Web Appendix 8A pressing ˇ 8 times, the y-intercept (a) will be shown, À8.92. Press ˇ again
and the slope coefficient (b) will be shown, 1.60. Press ˇ one more time and
the correlation coefficient, 0.91, will be shown.
Putting it all together, you should have the regression line shown here:
"J ¼ À 8:92 þ 1:60"M
r ¼ 0:91 Hewlett-Packard 10BII1
6. Press & C ALL to clear your memory registers.
Enter the first X value ("M ¼ 23.8 in our example) and press INPUT . Then
enter the first Y value ("J ¼ 38.6) and press
þ . Make sure you enter the X
Repeat Step 2 until all values have been entered.
To display the vertical axis intercept, press 0 & y ,m: À8.92 should appear.
To display the beta coefficient, b, press & SWAP . 1.60 should appear.
To obtain the correlation coefficient, press & x , r and then & SWAP to get
r ¼ 0:91. Putting it all together, you should have the regression line shown here:
"J ¼ À 8:92 þ 1:60 "M
r ¼ 0:91 Microsoft Excel
1. Manually enter into the spreadsheet the data for the market and for Stock J. 1
6 2. A
40.10% Access Microsoft Excel’s Regression tool from the Data Analysis package in
the Tools menu (Tools > Data Analysis > Regression). If you do not have the
Data Analysis package, you will have to add the Analysis ToolPak by
accessing Tools > Add-Ins.2
The dialog box that appears requires that you enter the Y and X variable
ranges and has additional options pertaining to what output is to be produced
and where it should be displayed. In this example regarding Stock J, the
“Input Y Range” prompt requires that Cells C2:C6 be entered as the dependent variable of the regression. Similarly, the “Input X Range” prompt
requires Cells B2:B6 as the independent variable. The Hewlett-Packard 17B calculator is even easier to use. If you have one, see Chapter 9 of the Owner’s Manual.
If you’re using Excel 2007, this step varies slightly. Access Excel’s regression tool by clicking “Data Analysis” under
the Data tab. Then using the arrow key to scroll down the dialog box, select “Regression” and click OK (Data >
Data Analysis > Regression > OK). The regression dialog box that follows is identical to the Microsoft Excel 97
version. If you do not have the Data Analysis package, you will have to add the Analysis ToolPak by clicking the
Microsoft Office button (which is at the left top corner), click the Excel Options tab, and then click Add-Ins. From
the Manage box, select “Excel Add-Ins” and click Go. From the next dialog box, select the Analysis ToolPak check
box and click OK.
2 8A-3 8A-4 Web Appendix 8A 3.
SUMMARY OUTPUT For the purposes of this example, none of the additional options are
chosen and the regression output relies on the default selection, which is
displayed on an additional worksheet.
Select “OK” to perform the indicated regression.
The following is a section of the output generated from the regression of Stock
J’s return on the market return. B C D E
15.103 Significance F
0.030195958 t Stat
0.0302 Lower 95%
0.29033775 G Regression Statistics
Adjusted R Square
X Variable 1 Coefficients
1.60309159 5. 6. Standard Error
0.412498347 Upper 95%
2.9158454 In this simple regression, the multiple R statistic is equivalent to the
correlation coefficient obtained in the other regression procedures described.
Hence, the correlation coefficient, r, is 0.91339175.
In the last section of the output, the intercept of the regression line is
–0.0892194 and the beta coefficient is 1.60309159. These results agree with
those obtained previously with financial calculators except that the intercept is
–0.089219 instead of –8.9219. The reason for this difference is that the returns
were entered as whole numbers in the calculator but were expressed as percentages in the spreadsheet. It is simply a matter of scale and does not have a
real effect on results.
The remainder of the output concerns the reliability of the estimates made and
is more fully explained in statistics courses. Web Appendix 8A Putting it all together, you should have the following regression line:
"J ¼ À 8:92 þ 1:60 "M
r ¼ 0:91 As illustrated, spreadsheet programs yield the same results as a calculator; however, the spreadsheet is more flexible and allows for a more thorough analysis.
First, the file can be retained; and when new data become available, they can be
added and a new beta can be calculated quite rapidly. Second, the regression
output can include graphs and statistical information designed to give us an idea
of how stable the beta coefficient is. In other words, while our beta was calculated
to be 1.60, the “true beta” might actually be higher or lower; the regression output
can give us an idea of how large the error might be. Third, the spreadsheet can be
used to calculate returns data from historical stock price and dividend information; then the returns can be fed into the regression routine to calculate the beta
coefficient. This is important because stock market data are generally provided in
the form of stock prices and dividends, making it necessary to calculate returns.
This can be a big job if a number of different companies and a number of time
periods are involved. PROBLEMS
8A-1 BETA COEFFICIENTS AND RATES OF RETURN You are given the following set of data: HISTORICAL RATES OF RETURN (")
Stock Y ("Y ) NYSE ("M )
c. d. Construct a scatter diagram graph (on graph paper) showing the relationship
between returns on Stock Y and the market, as in Figure 8A-1; then draw a freehand
approximation of the regression line. What is the approximate value of the beta
coefficient? (If you have a calculator with statistical functions, use it to calculate beta.)
Give a verbal interpretation of what the regression line and the beta coefficient show
about Stock Y’s volatility and relative riskiness as compared with other stocks.
Suppose the scatter of points had been more spread out but the regression line was
exactly where your present graph shows it. How would this affect (1) the firm’s risk
if the stock was held in a one-asset portfolio and (2) the actual risk premium on
the stock if the CAPM held exactly? How would the degree of scatter (or the correlation coefficient) affect your confidence that the calculated beta will hold true in the
Suppose the regression line had been downward-sloping and the beta coefficient had
been negative. What would this imply about (1) Stock Y’s relative riskiness and (2) its
probable risk premium? 8A-5 8A-6 Web Appendix 8A e. Construct an illustrative probability distribution graph of returns (see Figure 8-2)
for portfolios consisting of (1) only Stock Y, (2) 1% each of 100 stocks with beta
coefficients similar to that of Stock Y, and (3) all stocks (that is, the distribution of
returns on the market). Use as the expected rate of return the arithmetic mean given
previously for both Stock Y and the market and assume that the distributions are
normal. Are the expected returns “reasonable”—that is, is it reasonable that ^Y ¼ ^M ¼ 9:8%?
f. Now suppose that in the next year, Year 12, the market return was 27% but Firm Y
increased its use of debt, which raised its perceived risk to investors. Do you think
that the return on Stock Y in Year 12 could be approximated by this historical
characteristic line? Explain.
"Y ¼ 3:8% þ 0:62ð^M Þ ¼ 3:8% þ 0:62ð27%Þ ¼ 20:5%
r g. 8A-2 r
Now suppose "M in Year 12, after the debt ratio was increased, had actually been 0%.
What would the new beta be based on the most recent 11 years of data (that is, Years 2
through 12)? Does this beta seem reasonable—that is, is the change in beta consistent
with the other facts given in the problem? Explain. SECURITY MARKET LINE You are given the following historical data on market returns, "M , and the returns on Stocks A and B, "A and "B :
21.35 rRF, the risk-free rate, is 9%. Your probability distribution for rM for next year is as follows:
Probability rM 0.1
44 a. Determine graphically the beta coefficients for Stocks A and B. b. Graph the Security Market Line and give its equation. c. Calculate the required rates of return on Stocks A and B. d. Suppose a new stock, C, with ^C ¼ 18% and bC ¼ 2.0 becomes available. Is this stock
in equilibrium; that is, does the required rate of return on Stock C equal its expected
return? Explain. If the stock is not in equilibrium, explain how equilibrium will be
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