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Measuring the Beta using
Historical Stock Prices
2039
In this slide set
The beta coefficient
The linear regression approach to beta
measurement using historical return data
– Normalizing the data
– Normalized holding period returns
– Running the regression using MS Excel
– Relevant regression statistics and their
interpretation
– Different regression charts
The Beta Coefficient
Under the theory of the Capital Asset Pricing Model total risk is
partitioned into two parts:
– Systematic risk
– Unsystematic risk
Total Risk of the Investment
Systematic Risk Unsystematic Risk
Systematic risk is the only relevant risk to the diversified
investor
The beta coefficient measures systematic risk
The Term – “Relevant Risk”
What does the term “relevant risk” mean in the context of the CAPM?
– It is generally assumed that all investors are wealth maximizing
risk averse people
– It is also assumed that the markets where these people trade are
highly efficient
– In a highly efficient market, the prices of all the securities adjust
instantly to cause the expected return of the investment to equal
the required return
– When E(r) = R(r) then the market price of the stock equals its
inherent worth (intrinsic value)
– In this perfect world, the R(r) then will justly and appropriately
compensate the investor only for the risk that they perceive as
relevant…hence investors are only rewarded for systematic
risk…risk that can be diversified away IS…and prices and returns
reflect ONLY systematic risk.
The Proportion of Total Risk that is
Systematic
Each investor varies in the percentage of total risk that is
systematic
Some stocks have virtually no systematic risk.
– Such stocks are not influenced by the health of the economy in
general…their financial results are predominantly influenced by
company-specific factors
– An example is cigarette companies…people consume cigarettes
because they are addicted…so it doesn‟t matter whether the
economy is healthy or not…they just continue to smoke
Some stocks have a high proportion of their total risk that is
systematic
– Returns on these stocks are strongly influenced by the health of
the economy
– Durable goods manufacturers tend to have a high degree of
systematic risk
The Regression Approach to
Measuring the Beta
• You need to gather historical data about the stock and the market
• You can use annual data, monthly data, weekly data or daily data.
• You need at least thirty (30) observations of historical data.
• Hopefully, the period over which you study the historical returns of the
stock is representative of the normal condition of the firm and its
relationship to the market.
• If the firm has changed fundamentally since these data were produced
(for example, they have merged with another firm or have divested
itself of a major subsidiary) there is good reason to believe that future
returns will not reflect the past…and this approach to beta estimation
SHOULD NOT be used….rather, use the ex ante approach.
Historical Beta Estimation
In this example, we have determined the quarterly returns on the stock and the
market and using Excel…ran a regression to produce the accompanying chart.
Period HPR(Stock) HPR(TSE 300)
1990.1 -0.04 0.012
Characteristic Line (Regression)
0.3
1990.2 -0.16 -0.07
Returns on Stock
1990.3 0.32 0.12 0.2
1990.4 0.16 0.08
1991.1 -0.22 -0.11 0.1
1991.2 0.15 0.16
1991.3 0.28 0.13 0
1991.4 0.19 0.07 -0.5 0 0.5
1992.1 -0.16 -0.04 -0.1
1992.2 0.08 0.16
-0.2
1992.3 -0.03 -0.11
1992.4 0.34 0.25 Returns on TSE 300
Characteristic Line
The characteristic line is a regression line that represents the
relationship between the returns on the stock and the returns on the
market over a period of time.
The slope of the Characteristic Line is the Beta Coefficient
The degree to which the characteristic line explains the variability in the
dependent variable (returns on the stock) is measured by the
coefficient of determination. (also known as the R2 (r-squared or
coefficient of determination)).
If the coefficient of determination equals 1.00, this would mean that all
of the points of observation would lie on the line. This would mean that
the characteristic line would explain 100% of the variability of the
dependent variable.
The alpha is the vertical intercept of the regression (characteristic line).
Many stock analysts search out stocks with high alphas.
Characteristic Line for Imperial
Tobacco
Characteristic
Returns on
Line for Imperial
Imperial
Tobacco
Tobacco %
• High alpha
• R-square is
very low
• Beta is
irrelevant
Returns on
the Market %
(TSE 300)
High R2
An R2 that approaches 1.00 (or 100%) indicates that the
characteristic (regression) line explains virtually all of the
variability in the dependent variable.
This means that virtually of the risk of the security is
„systematic‟.
This also means that the regression model has a strong
predictive ability. … if you can predict what the market will
do…then you can predict the returns on the stock itself with a
great deal of accuracy.
Characteristic Line General Motors
Characteristic
Returns on
Line for GM
Imperial
Tobacco % (high R2)
• Positive alpha
• R-square is
very high
• Beta is
positive and
close to 1.0
Returns on
the Market %
(TSE 300)
An unusual Characteristic Line
Returns on a Characteristic Line for a stock
Stock % that will provide excellent
portfolio diversification
• Positive alpha
(high R2)
• R-square is
very high
• Beta is
negative and <
1.0
Returns on
the Market %
(TSE 300)
Diversifiable Risk
(non-systematic risk)
Examples of this type of risk include:
– a single company strike
– a spectacular innovation discovered through the company‟s R&D
program
– equipment failure for that one company
– management competence or management incompetence for that
particular firm
– a jet carrying the senior management team of the firm crashes
– the patented formula for a new drug discovered by the firm.
Obviously, diversifiable risk is that unique factor that influences
only the one firm.
OK – lets go back and look at raw data
gathering and data normalization
A common source for stock of information is Yahoo.com
You will also need to go to the library a use the TSE Review (a
monthly periodical)
You want data for at least 30 months.
For each month you will need:
– Ending stock price
– Number of shares outstanding for the stock
– Dividend per share paid during the month for the stock
– Ending value of the market indicator series you plan to use (ie.
TSE 300 composite index)
Demonstration Through Example
The following slides will
be based on Alcan
Aluminum (AL.TO)
Five Year Stock Price Chart for AL.TO
Spreadsheet Data From Yahoo
Process:
– Go to http://ca.finance.yahoo.com
– Use the symbol lookup function to search for the
company you are interested in studying
– Use the historical quotes button…and get 30
months of historical data
– Use the download in spreadsheet format feature
to save the data to your harddrive
Spreadsheet Data From Yahoo
The raw downloaded data should look like this:
Date Open High Low Close Volume
01-May-02 57.46 62.39 56.61 59.22 753874
01-Apr-02 62.9 63.61 56.25 57.9 879210
01-Mar-02 64.9 66.81 61.68 63.03 974368
01-Feb-02 61.65 65.67 58.75 64.86 836373
02-Jan-02 57.15 62.37 54.93 61.85 989030
03-Dec-01 56.6 60.49 55.2 57.15 833280
01-Nov-01 49 58.02 47.08 56.69 779509
Spreadsheet Data From Yahoo
The raw downloaded data should look like this:
Date Open High Low Close Volume
01-May-02 57.46 62.39 56.61 59.22 753874
01-Apr-02 62.9 63.61 56.25 57.9 879210
Volume of
Opening price per share, the trading done
The day, highest price per share during the in the stock
month and month, the lowest price per share on the TSE in
year achieved during the month and the the month in
closing price per share at the end numbers of
of the month board lots
Spreadsheet Data From Yahoo
From Yahoo, the only information you can use is the
closing price per share and the date. Just delete the
other columns.
Date Close
01-May-02 59.22
01-Apr-02 57.9
01-Mar-02 63.03
01-Feb-02 64.86
02-Jan-02 61.85
Acquiring the Additional Information
You Need
In addition to the closing price of the stock on a per share basis,
you will need to find out how many shares were outstanding at
the end of the month and whether any dividends were paid
during the month.
You will also want to find the end-of-the-month value of the
S&P/TSX Total Return Composite Index (look in the green
pages)
You will find all of this in The TSE Review periodicals (HG
5160.T6T6) found on the second floor of the library.
Raw Company Data
Closing Price Cash
Issued for Alcan Dividends
Date Capital AL.TO per Share
01-May-02 321,400,589 $59.22 $0.00
01-Apr-02 321,400,589 $57.90 $0.15
01-Mar-02 321,400,589 $63.03 $0.00
01-Feb-02 321,400,589 $64.86 $0.00
02-Jan-02 160,700,295 $123.70 $0.30
01-Dec-01 160,700,295 $119.30 $0.00
Number of shares doubled and share price fell in half
– this is indicative of a 2 for 1 stock split.
Normalizing the Raw Company Data
Closing Cash
Issued Price for Dividends Adjustment Normalized Normalized
Date Capital Alcan per Share Factor Stock Price Dividend
01-May-02 321,400,589 $59.22 $0.00 1.00 $59.22 $0.00
01-Apr-02 321,400,589 $57.90 $0.15 1.00 $57.90 $0.15
01-Mar-02 321,400,589 $63.03 $0.00 1.00 $63.03 $0.00
01-Feb-02 321,400,589 $64.86 $0.00 1.00 $64.86 $0.00
02-Jan-02 160,700,295 $123.70 $0.30 0.50 $61.85 $0.15
01-Dec-01 145,000,500 $111.40 $0.00 0.45 $50.26 $0.00
The adjustment factor is just the value in the issued
capital cell dividend by 321,400,589.
Calculating the HPR on the stock from
the normalized data
Normalized Normalized
Date Stock Price Dividend HPR
( P P0 ) D1
HPR
01-May-02 $59.22 $0.00 2.28% 1
01-Apr-02 $57.90 $0.15 -7.90%
01-Mar-02 $63.03 $0.00 -2.82%
P0
01-Feb-02 $64.86 $0.00 4.87%
02-Jan-02 $61.85 $0.15 23.36%
01-Dec-01 $50.26 $0.00
Use $59.22 as the ending price, $57.90 as the
beginning price and during the month of May, no
dividend was declared.
Now Put the data from the S&P/TSX
Total Return Composite Index in
Ending
Normalized Normalized TSX
Date Stock Price Dividend HPR Value
01-May-02 $59.22 $0.00 2.28% 16911.33
01-Apr-02 $57.90 $0.15 -7.90% 16903.36
01-Mar-02 $63.03 $0.00 -2.82% 17308.41
01-Feb-02 $64.86 $0.00 4.87% 16801.82
02-Jan-02 $61.85 $0.15 23.36% 16908.11
01-Dec-01 $50.26 $0.00 16881.75
You will find the Total Return S&P/TSX Composite
Index values in TSE Review found in the library.
Now Calculate the HPR on the Market
Index
Ending
Normalized Normalized TSX HPR on
Date Stock Price Dividend HPR Value the TSX
01-May-02 $59.22 $0.00 2.28% 16911.33 0.05%
01-Apr-02 $57.90 $0.15 -7.90% 16903.36 -2.34%
01-Mar-02 $63.03 $0.00 -2.82% 17308.41 3.02%
01-Feb-02 $64.86 $0.00 4.87% 16801.82 -0.63%
02-Jan-02 $61.85 $0.15 23.36% 16908.11 0.16%
01-Dec-01 $50.26 $0.00 16881.75
Again, you simply use the HPR formula using the
ending values for the total return composite index.
Regression In Excel
If you haven‟t already…go to the tools
menu…down to add-ins and check off the
VBA Analysis Pac
When you go back to the tools menu, you
should now find the Data Analysis bar, under
that find regression, define your dependent
and independent variable ranges, your
output range and run the regression.
Now Use the Regression Function in
Excel to regress the returns of the
stock against the returns of the market
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.05300947
R Square 0.00281 R-square = coefficient of
Adjusted R Square -0.2464875
Standard Error 5.79609628
determination
Observations 6
ANOVA
df SS MS F Significance F
Regression 1 0.3786694 0.37866937 0.011271689 0.920560274
Residual 4 134.37893 33.5947321
Total 5 134.7576
CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0%
Intercept 59.3420816 2.8980481 20.4765686 3.3593E-05 51.29579335 67.38836984 51.2957934 67.38837
X Variable 1 3.55278937 33.463777 0.10616821 0.920560274 -89.35774428 96.46332302 -89.3577443 96.46332
Alpha
Beta
Finalize Your Chart
You can use the charting feature in Excel to
create a scatter plot of the points and to put a
line of best fit (the characteristic line) through
the points.
Finally, you will want to interpret the Beta (X-
coefficient) the alpha (vertical intercept) and
the coefficient of determination.
The Beta
Obviously the beta (X-coefficient) can simply
be read from the regression output.
You will want to interpret it in the context of
the firms, its products and the likely
relationship that they hold with the health of
the overall market.
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